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STAIR-BUILDING 

IN  ITS  VARIOUS  FORMS; 

AND 

The  New  ONE-PLA^|,E,;.]vf;ETffQi). 


AS  APPLIED  TO 


Drawing  Face-Moulds, 
Unfolding  the  Centre  Line  of  Wreaths, 


THEREBY 


OBTAINING  EXACT  LENGTHS  OF  BALUSTERS, 
AND  ALSO  UNFOLDING  SIDE  MOULDS. 

111I10J1S  BISMS  111  PLUS  OF  STIua  IIWIIS  111  MMMhimm. 

FOR  THE  USE  OF 

ARCHITECTS, 

Stair-Builders,  Carpenters,  Iron- Workers,  Pattern- Makers,  and 

STONE  MASONS. 

WOOD,  mON,  AND  STONE  STAIRS. 


By  JAMES  H.  MONCKTON, 

Author  of  the  '^American  Stair-Builder,"    Monckton' s  National  Carpenter  and  Joiner"  "  Monckton  s  Nationa^ 
Stair-Builder  "  and  "  Monckton' s  Practical  Geometry" ;  Teacher  for  many  years  of  the 
Mechanical  Class  in  the  "  General  Society  of  Mechanics  and  Tradesmeti's 
Free  Drawing  School  of  the  City  of  New  York." 


FOURTH  EDITION,  REVISED  AND  EXTENSIVELY  ENLARGED. 


NEW  YORK : 
JOHN   WILEY   &  SONS, 
53  East  Tenth  Street. 
1894. 


t'  /S 


COPVRIGHT,  1894. 

By  JAMES  H.  MONCKTON. 


PREFACE  TO  FOURTH  EDITION. 


NoTwi  fHSTANDiNG  our  former  belief  in  the  completeness  of  this  work  at  the  conclusion  of  its  preparation 
and  first  publication,  we  have  found  since  that  time,  from  our  own  careful  working  experience  and  study, 
added  to  the  many  practical  questions  asked  by  interested  and  intelligent  correspondents  in  different  por- 
tions of  the  world,  that  it  was  important  to  make  many  changes  and  additions.  Numerous  revisions  and 
special  changes  have  been  made,  all  intended  to  perfect,  widen,  and  maintain  the  usefulness  and  raise  the 
scientific  and  practical  character  of  the  work.  At  this  time  and  opportunity  of  expression  to  our  readers, 
we  have  only  to  reaffirm  that  Geometry  as  applied  to  Stair  Building  and  Hand-Railing  throughout  this  book 
is  of  the  simplest  character  in  its  methods  and  perfect  in  its  practical  results.  Some  sixteen  entirely  new 
plates  have  been  added  that  have  extended  uses  in  the  practice  of  building  Wood,  Iron,  and  Stone  Stairs. 
We  will  briefly  mention  some  of  the  foremost  useful  additions  that  were  never  before  published : — Plate  73. 
Wood  Panelling  of  the  Warped  Soffit  of  Circular  Stairs.  Plate  76.  Side  Moulds  unfolded  geometrically 
correct  by  the  use  of  a  level  line  common  to  both  planes.  (The  geometry  of  this  unfoldment  we  have  pub- 
lished before  in  unfolding  a  central  wreath-line  for  finding  lengths  of  balusters,  etc.)  Cutting  Square-top 
Balusters  to  fit  the  Warped  Surface  of  Wreathed  Hand-rail.  Plate  79.  Hand-rail  in  One  Piece  over 
Cylinders  of  Platform  Stairs.  Plate  80.  Circular  Stone-Work.  Plate  82.  Marble  Wainscoting  of  a 
Circular  Staircase,  treated  with  Heavy  Cap  and  Base  Mouldings.  Plate  83.  Unfoldment  of  the  Warped 
Panelled  Soffit  of  a  Stone  Circular  Staircase.  Plate  84.  Hand-rail  of  Circular  Stone  Stairs.  Novel  Plans 
and  Elevations  of  Iron  Stairs. 

JAS.  H.  MONCKTON. 


PREFACE. 


This  book  presents  for  the  first  time— as  applied  to  hand-railing— ^Z^^-  07ie-plane  method  of  draw- 
ing face-moulds,  a  method  in  which  only  one  plane  of  projection  is  required;  the  simplicity,  rapidity 
and  convenience  of  which  in  practice  make  it  superior  to  all  others.  By  the  one-plane  process  of 
drawing  a  face-mould,  a  centre  line  of  wreath  may  also  be  unfolded  and  fixed  in  its  relation  to  the 
elevation  of  tread  and  rise;  thus  determining  the  length  of  each  baluster  on  the  curved  plan  and  gain- 
ing a  knowledge  and  control  of  the  wreath's  exact  position  not  before  attainable. 

It  is  intended  to  make  this  book  a  complete  work  on  stair-building  and  hand-railing,  giving  a 
large  selection  of  plans  and  designs  of  staircases,  newels  and  balusters,  and  numerous  examples  of 
construction.  The  paper  solids  here  introduced  as  an  objective  means  of  elementary  and  practical 
instruction  in  the  principles  and  geometrical  methods  of  hand-railing  are  unequalled  for  the  pur- 
pose, as  the  construction  of  these  solids  with  the  drawings  on  their  surfaces  show  all  the  positions 
and  connections  required  in  each  case,  and  thereby  enable  any  fairly  intelligent  person  to  understand 
and  grasp  the  subject.  The  professional  architect  will  find  valuable  suggestions  in  design  and 
construction  ;  also  important  considerations  in  planning  stairs. 

The  experienced  stair-builder  will  learn  that  this  one-plane  method  of  drawing  all  face-moulds — 
and  also  the  manner  of  finding  the  angles  with  which  to  square  wreath-pieces — is  simple,  uniform  and 
rapid  ;  and  no  matter  how  skilful  a  stair-builder  he  may  be,  he  will  find  that,  in  the  extent  and  com- 
pleteness of  detail  with  which  the  subject  is  treated,  it  will  prove  a  valuable  work  of  reference.  The 
expert  rail-worker  will  learn  of  the  geometrical  law  controlling  the  top  and  bottom  curves  of  every 
wreath-piece,  showing  that  a  face-mould  is  not  only  a  means  of  shaping  the  sides  of  a  wreath-piece  on 
the  plane  of  the  plank,  but  that  it  carries  with  it  a  central  geometrical  curve  (a  helical  line)  that  must 
be  observed  in  shaping  the  top  and  bottom  surfaces  of  the  wreath.  To  prove  this  in  a  practical  way 
it  is  only  necessary  to  call  attention  to  the  fact  that  in  the  case  of  round  hand-rail  over  any  curved 
plan,  its  sides  hang  vertically  over  the  plan,  while  its  top  and  bottom  form  proper  curves  giving  its 
own  easings  perfectly  suited  to  the  requirements  in  all  cases  ;  and  when  it  is  considered  that  a 
moulded  rail  over  the  same  plan  would  be  subject  to  the  same  centre  and  tangents,  with  the  same 
joints,  then  the  absolute  control  of  the  curves  forming  the  top  and  the  bottom  of  a  wreath  by  this 
central.geometrical  curve-line  becomes  self-evident.^  In  connection  with  the  above  statements  examine 
for  instance  Plate  48,  as  giving  one  example  among  many  of  the  correctness  and  practical  value  of  un- 
folding the  centre  line  of  a  wreath. 

The  student  or  apprentice  will  find  that  the  elementary  study  of  hand-railing  in  the  practical  and 
novel  way  here  presented  is  easily  acquired  ;  he  will  also  see  that  the  detail  instruction  given  in  stair- 
building  from  a  stepladder  to  expensive  and  difficult  staircases  is  presented  in  a  manner  to  be  clearly 
understood  and  quickly  learned.  That  the  drawings  and  descriptive  page  should  be  opposite  I  regard 
as  of  no  slight  importance  in  a  work  of  this  character  ;  those  who  have  experienced  the  weary  task  of 
turning  from  reference  pages  to  plates  located  at  another  portion  of  the  book  will  appreciate  the  value 
of  this  arrangement.  The  comprehensive  system  of  hand-railing  here  given  covers  every  practical  re- 
quirement, as  follows:  ist.  By  the  use  of  tangents  controlling  the  inclination  of  the  plane  of  the  plank 
and  the  butt-joints  ;  2d.  By  the  one-plane  method  of  drawing  all  face-moulds,  simply  applying  to  this 
purpose  a  level  line  common  to  both  planes  ;  3d.  By  the  further  use  of  this  last-mentioned  level  line,  in 
unfolding  the  centre  line  of  wreath,  which  also  unfolds  correct  forms  of  side  moulds  ; — all  of  which  are  based 
upon  the  demonstrable  laws  of  geometry,  and  point  to  a  conclusion  in  the  science  of  hand-railing  as  plain  as 
that  in  the  decimal  system  of  numbers,  which,  based  upon  the  ever-truthful  laws  of  that  brancii  of  mathe- 
matics, is  simple  in  its  methods  and  perfect  in  its  results. 

The  instruction  and  drawings  tliroughout  this  book  apply  ec[ually — except  in  some  minor  details,  ivhich 
special  drawings  supply — to  Wood,  Iron,  and  Stone  Stairs,  including  Hand  Railing. 

Finally,  it  is  my  belief  that  this  work  is  carried  to  an  extent  far  beyond  any  publication  that  has 
preceded  it  ;  practically  and  scientifically  covering  the  whole  field  of  stair-building  and  hand-railing, 
making  a  complete  digest  of  the  subject. 

Jas.  H.  Monckton. 


CONTENTS. 


HISTORY  OF  STAIRS 

PAGE 

With  seven  engravings,  examples  of  ancient  English  stairs  xxi,  xxii,  xxiii^  xxiv 

DEFINITIONS 

Of  terms  used  in  connection  witli  stairs  and  stair-building  xxv  and  xxvi 

LIST  OF  BOOKS 

Published  in  England  and  America,  treating  partially  or  wholly  on  stairs  or  hand-railing,     .       .       .  xxvii 

SUGGESTIONS. 

To  teachers  of  technical  schools,   xxix 

To  apprentices  and  students,   xxix 

Concerning  the  study  of  hand-railing,        .   xxix 

Advantage  of  squaring  model  wreath-pieces,   xxix 

Fitting  wreaths  over  circular  or  other  curved-iron  staircases,   xxix 

As  to  the  fitness  of  close-paneled  strings,          .............  xxix 

Self-supporting  circular  stairs,      .   xxix 

Joints  of  wreath -pieces,  .   xxix 

Turned  newels,   xxix 

Connecting  hand-rails  with  newels,   xxix 

Balusters  with  square  bases,   xxix 

Loose  newel-caps  mitred  to  hand-rails,       ..............  xxix 

Setting  up  hand-rails  finished  and  varnished,   xxix 

IRON  STAIRS. 

Scientific  and  artistic  treatment  of  iron  stairs,  ............. 

Designs  and  Plans  for  Iron  Stairs  Plates  86  to  104 

Circular  Stone  Work  Plate  80 

STONE  STAIRS. 

Circular  Stone  Stairs,  etc  Plates  81  1085 


PLANS  AND  ELEVATIONS. 


PLATE  1. 

Plan  and  elevation  of  a  straight  flight  of  stairs  with  a  7"  cylinder.  Plan  and  elevation  of  a  half-turn 
platform  stairs  with  a  6"  cylinder,  landing  with  four  rises  above  the  platform.  A  rule  to  find  the  correct 
proportion  of  tread  to  rise. 

PLATE  2. 
STEP-LADDERS  AND  STOOP. 

Plans  of  step-ladders.  Elevations  of  step-ladders.  Elevation  of  stoop  with  platform,  hand-rail,  balusters 
and  newel. 

PLATE  3. 

T/ie  old  English  method  of  stair-bitildinff.  Finishing  stairs  on  the  under-side,  showing  the  construction. 
Ptittina-  lep  stairs.  Covering  stairs.  Stair-timbering  and  rough-bracl<eting.  Best  method  of  joining  hand- 
rail to  newel.  J^/an  and  elevation  of  a  common  straight  flight  of  stairs.  Methods  of  making  ordinary 
small  cylindeis  and  splicing  them  to  strings.  Front  and  wall  strings  laid  out.  Step  and  riser  as  worked 
and  glued  together. 

PLATE  4. 

PLANNING  WINDING  STAIRS. 

Line  of  travel  fi.xed.  Elevation  of  the  winding  flight  given.  Laying  out  the  front-string  and  cylinders. 
Laying  out  the  wall-strings. 

PLATE  5. 
PLANNING  STAIRS. 

In  the  following  three  Plates — 5,  6  and  7 — over  thirty  different  plans  of  stairs  are  given  ;  their  various 
dimensions  are  figured  for  the  convenience  of  examination  and  use  in  the  study  and  planning  of  stairs. 
Also  reference  is  given  to  each  Plate,  where  a  drawing  of  the  plan  of  stairs  is  again  projected  on  a  large 
scale,  together  with  the  management  of  the  hand-rail  wreath. 

Fig.  1.  Plan  of  a  straight  flight  of  stairs  starting  and  landing  with  small  cylinders.  See  Plate  i, 
Figs,  i  and  2;  also  Plates  3  and  22. 

Fig.  2.  Plan  of  a  straight  flight  of  stairs  starting  with  a  newel  set  off  2J",  and  landing  with  a  7" 
cylinder,  the  landing  riser  set  into  the  cylinder  2\".     See  Plate  30,   FiGS-  i  and  2:  also  Plates  33  and  34. 

Fig.  3.  Plan  of  winding  stairs  suitable  for  Basement  or  Attic  Story,  with  mortised — or  housed — strings 
both  sides ;  intended  to  be  set  between  partitions  or  otherwise  enclosed. 

Fig.  4.  Shows  the  manner  of  laying  out  the  front  string.  A  and  B  are  continuations  of  the  string,  5" 
wide,  and  spliced  to  the  string  as  shown. 

Fig.  5.  Plan  of  a  qtiarter-turn  of  winders,  with  cylinder,  suitable  for  either  the  starting  or  landing  of  a 
flight  of  stairs.    See  Plate  25. 

Fig.  6.  A  much  superior  plan  to  that  of  FiG.  5,  turning  one  quarter  without  winders,  and  with  the  same 
size  of  cylinder  and  regular  tread.  In  this  plan,  by  curving  the  front  ends  of  two  risers,  winders  are  dispensed 
with  and  a  platform  and  parallel  steps  substituted  ;  this  plan  as  shown  at  D  C  requiring  7^"  more  run.  See 
Plate  26. 

Fig.  7.  Plan  the  same  as  FiG.  6,  but  instead  of  a  continued  hand  rail  a  small  newel  and  connecting  level 
quarter-cylinder  are  substituted.    See  Plate  62. 

Fig.  8.  Plan  for  starting  or  landing  a  flight  of  stairs,  with  a  single  newel  set  diagonally. 

Fig.  9.  Plan  of  stairs  combining  two  platforms  with  a  parallel  step  between  (and  the  front  ends  of  its 
two  risers  curved),  making  a  half-turn.    See  Plate  41. 

Fig.  10.  Plan  of  a  quarter-platform  stairs,  showing  how  to  place  the  risers  connecting  with  any  quarter- 
cylinder  so  that  the  wreath  may  be  worked  in  one  piece  on  a  common  inclination  and  in  the  best  possible 
shape.  On  these  plans,  from  the  face  of  the  riser  F  and  on  the  centre  line  of  the  rail,  F  E  should  equal  half 
a  tread ;  and  from  the  face  of  the  riser  G  on  the  centre  line  of  rail,  G  E  must  equal  half  a  tread.  See  Plate 
31.  Fig.  3,  and  Plate  37,  Figs.  5,  6  and  7. 

Fig.  11.    Plan  of  stairs,  with  a  quarter-cylinder  turning  one  quarter,  with  winders.    See  Plate  36. 

PLATE  6. 
PLANNING  STAIRS. 

Fig.  1.  Plan  of  half-turn  platform  stairs  with  the  risers  set  in  the  cylinder  as  far  as  possible  to  save  run. 
The  cylinder-opening  is  made  up  of  two  6"  quarter-circles  and  5"  straight  between.    See  Plate  38. 


xii 


CONTENTS. 


Fig.  2.  Plan  of  a  flight  of  winding  stairs  with  lo"  cylinders,  making  a  half-turn  at  the  starting  and  at  the 
landing.    See  Plates  28  and  29. 

Fig.  3.  Plan  of  two  connecting  flights  starting  and  landing  from  one  1 2'' cylinder ,  the  flight  starting 
tiiriiing  one  quarter  with  winders,  the  other  flight  straight  and  landing  straigiit,  with  its  top  riser  set  into 
the  cylinder  4".    See  Plate  42. 

Fig.  4.  Plan  of  a  landing  or  starting  of  a  flight  of  stairs,  with  a  single  newel  set  between  quarter-cylinders. 
See  Plate  37,  Figs,  i,  2,  3  and  4. 

Fig.  5.  Plan  of  the  starting  of  a  flight  of  stairs,  with  the  front-string  curved  out.  including  four  treads 
Each  of  the  four  treads  are  increased  in  width  at  the  wall-string  and  diinmislied  at  the  front-string, 
curving  the  risers, — making  what  are  called  swell  steps.     See  Plate  32. 

Fig.  6.  Plan  of  the  starting  or  landing  of  a  flight,  turning  one  quarter,  with  platform  ;  finished  with  low- 
down  newels,  or — more  properly — small  corner-i)ieces,  and  continued  hand  rail  with  ramp  and  goose-neck.  See 
Plate  58. 

Fig.  7.    Plan  of  half-turn  plaiform  open-newel  stairs  with  wing-flights. 

Fig.  8.  Plan  of  half-turn  ])latforin  stairs  with  a  large  cylinder  into  which  the  front  ends  of  the  risers  of 
parallel  steps  are  brought  curved;  thus  ulihzing  the  cylinder-space  anfi  making  a  half-turn  without  resorting 
to  winders.    See  Plate  51. 

Fig.  9.  Plan  of  ship  or  s:''amboat  stairs.  At  Plate  52  is  a  plan  similar  to  this  with  angle  newels.  See 
also  Plate  50. 

PLATE  7. 
PLANNING  STAIRS. 

Fig.  1.  Plan  of  half-turn  platform-stairs  with  a  15"  cylinder;  the  risers  at  the  platform  and  at  the  starting 
of  the  flight  are  set  in  the  cylinders  6",  tliereby  saving  I'.o"  of  run.    See  Plate  40. 

Fig.  2.  Plan  of  a  half-turn  platform  stairs  wiili  a  central  riser  making  two  platforms,  and  the  starting 
and  landing  risers  each  set  into  the  cylinder  3".    See  Plate  39. 

Fig.  3.  Plan  of  a  flight  of  stairs  winding  one  quarter  at  the  starting  and  one  half  at  about  the  middle 
of  the  flight.  For  the  treatment  of  the  iiand-rail  at  the  starting  see  Plate  48,  and  for  the  treatment  of  the 
rail  at  the  half-turn  see  Plate  47. 

Fig.  4.  Plan  of  the  bottom  or  top  portion  of  a  flight  of  stairs  making  a  quarter-turn  with  a  quarter- 
platform,  the  platforvi  and  parallel  steps  secured  a?id  winders  avoided,  by  curving  the  front  ends  of  risers 
to  the  widths  of  treads  marked  on  the  line  of  cylinder.  The  cylinder  f)pening  is  formed  of  two  quarter- 
circles,  each  of  5''  radius  with  8'  slraiglit  between.  For  treatment  of  hand-rail  with  larger  scale-drawing 
see  Plate  46. 

Fig.  5.  Plan  of  the  bottom  or  top  portion  of  a  flight  of  open-newel  stairs  with  a  15"  opening,  making 
a  quarter-turn  with  a  quarter-platform.    See  Plate  59. 

Fig.  6.  Plan  of  the  bottom  or  to,)  portion  of  a  flight  of  stairs  with  a  10"  cylinder  winding  one 
quarter.  For  treatment  of  wreath  at  the  bottom  of  a  flight  see  Plate  27,  and  for  the  top  of  a  flight 
the  same  as  at  Plate  25. 

Fig.  7.  Plan  similar  to  Fu;.  6,  with  one  more  winder  in  the  plan  and  curved  risers  introduced  to 
sive  room  between  the  points  B  and  C. 

Fig.  8.  Plan  same  as  FiG.  4,  with  a  complete  semicircle  for  cylinder  opening.  For  treatment  of 
wreath  see  Plate  44. 

Fig.  9.  Plan  of  the  top  or  bottom  portion  of  a  flight  of  stairs  making  a  quarter  turn  with  a  quarter- 
platform  and  a  parallel  step  set  at  the  centre  of  a  15"  cylinder.  See  Plate  43  for  larger  size  scale- 
drawing  and  management  of  wreath. 

Fig.  10.  Plan  of  a  flight  of  circular  stairs.  The  dotted  lim  s  show  the  best  method  of  placing  the 
carriage-timbers.  The  treatment  of  hand-rail  on  a  larger  scale  of  drawing  will  be  found  at  Plate  53. 
At  Plate  54  full  instruction  is  given  for  changing  the  plan  of  the  first  step  to  the  scroll  form  ;  the  treat 
meat  of  that  portion  of  the  hand-rail,  also  the  construction  of  the  scroll-step. 

Fig.  11.  Plan  of  an  elliptic  flight  of  stairs  with  the  treads  at  the  wall  and  front-strings  properly 
graded  so  as  to  bring  the  risers  on  lines  more  nearly  normal  to  the  curves  and  at  the  same  time  main- 
tain an  even  tread  on  the  line  of  travel.    These  stairs  should  be  timbered  in  the  manner  shown  at  FiG.  10. 

Self-supporting  stairs,  also  see  Plate  45. 

PLATE  8. 

Figs.  1  and  2.    Bending  wood  by  saw-kerfing  and  the  construction  of  a  circular  form,  with  ribs  of  the 
proper  curve  an;l  narrow  strips  of  the  required  length  called  laggings,  over  which  to  bend  the  saw-kerfed  materiaL 
Fig.  3.    Rending  wood  on  a  curved  form  and  keying. 

Fig.  4.  Bending  veneer,  facing  and  fillmg  out  the  thickness  with  parallel  pieces  — fitted  to  the  curve — 
called  staves. 

Fig.  5.    Laminated   work  —bending  and  gluing  several  thickness  of  veneer  together.     A  rule  to  ascertain 
what  thickness  of  white  pine  will  bear  bending  any  given  radius  of  cur7'ature  without  injuring  its  elasticity. 
Fig.  6.    Bending  stair-strings. 

Fig3.  7  and  8.  Construction  of  a  form  for  bending  quarter-circle  stair-strings,  the  ribs  to  stand  on  an 
angle  parallel  to  the  inclination  of  string. 

Figs.  9  and  10.    Soffit-mouldings  on  the  lower  edge  of  cylinders. 

Figs.  11  and  12.    To  find  the  lengths  of  cylinder-staves  that  include  winder-treads. 

PLATE  9. 


Examples  of  the  one-plane  method  in  drawing  face  moulds.     Face-moulds,  their  number  and  character. 


CONTENTS. 


xiii 


ELEMENTARY  INSTRUCTION  IN  HAND-RAILING  OVER 

CURVED  PLANS. 

PLATE  10. 

Face-mould  and  parallel  pattern  over  a  plan  of  a  quarter-circle ;  one  tangent  inclined,  the  other  horizontal. 
The  angles  with  which  to  square  the  wreath-piece  at  the  joints. 

PLATE  11. 

Face-mould  and  parallel  pattern  over  a  plan  of  a  quarter-circle,  the  tangents  inclined  alike.  The  angle 
with  which  to  square  the  wreath-piece  at  both  joints. 

PLATE  12. 

Face-mould  and  parallel  pattern  over  a  plan  of  a  quarter-circle,  the  tangents  dilTerently  inclined.  The 
angles  with  which  to  square  the  wreath-piece  at  the  joints. 

PLATE  13. 

Face-mould  and  parallel  pattern  over  a  plan  less  than  a  quarter-circle,  one  tangent  inclined,  the  other 
horizontal.    The  angles  with  which  to  square  the  wreath-piece  at  the  joints. 

PLATE  14. 

Face-mould  and  parallel  pattern  over  a  plan  of  more  than  a  quarter-circle,  one  tangent  inclined,  the  other 
horizontal.    The  angles  with  which  to  square  the  wreath-piece  at  the  joints. 

PLATE  15. 

Face-mould  over  a  plan  less  than  a  quarter-circle,  the  tangents  inclined  alike.  The  angle  with  which  to 
square  the  wreath-piece  at  both  joints. 

PLATE  16. 

Face  n-ould  over  a  plan  less  than  a  quarter-circle,  the  tangents  differently  inclined.  The  angles  wiili 
which  to  square  the  wreath-piece  at  the  joints. 

PLATE  17. 

Face-mould  over  an  elliptic  or  eccentric  curved  plan,  the  tangents  of  unequal  length  and  different  inclina- 
tions. The  angles  with  which  to  square  the  wreath-piece  at  the  joints.  To  find  a  coiniiion  angle  of  inclina- 
tion over  two  differ e)it  lengths  of  pla7i  tangents — when  required — the  total  height  being  given. 

PLATE  18. 

Face  mould  or  parallel  pattern  over  a  plan  greater  than  a  quarter-circle,  the  tangents  mclined  alike.  The 
angle  with  which  to  square  the  wreath-piece  at  both  joints. 

PLATE  19. 

Face-mould  over  a  plan  greater  than  a  quarter-circle,  the  tangents  differently  inclined.  The  angles  with 
which  to  square  the  wreath-piece  at  the  joints. 


DEVELOPMENT  OR  UNFOLDING  OF  THE  CENTRE  LINE  OF 

WREATHS. 

PLATE  20. 

Fig.  1.  Plan  of  a  centre  line  of  wreath,  a  semicircle  with  three  tangents  inclined  alike,  the  fourth  tangent 
level. 

Fig.  2.  The  centre  line  of  wreath  unfolded  from  plan  Fig.  i. 

Fig.  3.  Plan  of  a  centre  line  of  wreath,  a  semicircle  with  two  tangents  inclined  alike,  the  third  of  a  less 
inclination,  and  the  fourth  tangent  level. 

Fig.  4.  The  centre  line  of  wreath  unfolded  from  plan  Fig.  3. 

Fig.  5.  Plan  of  a  centre  line  of  wreath-piece  less  than  a  quarter-circle,  one  tangent  inchned,  tlie  other 
level. 

Fig.  6.  The  centre  line  of  wreath-piece  unfolded  from  plan  Fig.  5. 


xiv 


CONTENTS. 


PLATE  21. 

Pig.  1.  Plan  of  a  centre  line  of  wreath-piece  greater  tlian  a  quarter-circle,  one  tangent  inclined,  tlie  other 
level. 

Fig.  2.  The  centre  line  of  wreath-piece  unfolded  from  plan  Fio.  i. 

Fig.  3.  Plan  of  a  centre  line  of  wreath-piece  greater  than  a  quarter-circle,  the  tangents  of  a  like 
inclination. 

Fig.  4.  The  centre  line  of  wreath-piece  unfolded  from  plan  FiG.  3. 

Fig.  5.  Plan  of  centre  line  of  wreath-piece  greater  than  a  quarter-circle,  the  tangents  diflerently 
inclined. 

Fig.  6.  The  centre  line  of  wreath-piece  unfolded  from  plan  FiG.  5. 

Fig.  7.  Plan  of  a  centre  line  of  wreath  piece  less  than  a  quarter-circle,  the  tangents  inclined  alike. 
Fig.  8.  The  centre  line  of  wreath-piece  unfolded  from  plan  Fig.  7. 

Fig.  9.  Plan  of  a  centre  line  of  wreath-piece  less  than  a  quarter-circle,  the  tangents  inclined  diflerently. 
Fig.  10.  The  centre  line  of  wreath-piece  unfolded  from  plan  Fig.  9. 


PLATE  22. 

POSITION  OF  RISERS  IN  CONNECTION  WITH  CYLINDERS  AT  THE  STARTING  AND  AT  THE 

LANDING  OF  STRAIGHT  FLIGHTS  OF  STAIRS. 

Fig.  1.  Elevation  of  rises  and  tread  to  determine  the  face  of  riser  at  the  bottom  of  a  flight  when  the 
over  \vo(;d  is  to  be  all  removed  from  the  top  of  the  wreath-piece. 

Fig.  2.  Elevation  of  tread  and  rise  at  the  top  of  a  flight  to  determine — 'when  all  the  over-wood  is  removed 
from  I  he  bottom  of  a  ivreath-piece — the  relative  position  of  the  riser  and  cylinder. 

Fig.  3.  Plan  of  the  bottom  of  flight  with  riser  and  cylinder  placed  as  determined  at  FiG.  i. 

Fig.  4.  Plan  of  the  top  c;f  flight  with  cylinder  and  riser  as  determined  by  trial  at  FiG.  2. 

Fig.  5.  Face-mould. 

Fig.  6.  Perspective  sketch  of  the  wreath-piece,  showing  both  joints  prepared  for  squaring  and  the  applica- 
tion of  the  face-mould  both  sides  of  the  plank. 

Fig.  7.  Elevation  of  tread  and  rises  for  the  top  and  bottom  of  flight,  showing  that  in  some  cases  by  a 
change  in  removing  the  over-wood  the  risers  may  be  placed  at  the  chord-lines  of  cylinders  of  this  diameter. 

Laying  out  the  joint  with  the  pitch-board  as  at  FiG.  5  shows  how  to  And  the  thickness  of  wood  required  for 
my  size  and  form  of  hand-rail. 


PLATE  23. 

Half-turn  Platform  Stairs  as  given  by  plan  and  elevation  at  Plate  i.  Figs.  3  and  4.  Ho'm  to  fix 
the  position  of  risers  in  connection  with  the  cvliudn  s,  and  the  changes  possible  by  varying  tlie  removal  of 
;)ver-wood  from  the  straight  portion  of  wi cii h-[ licces.  see  FiGS.  i  and  2;  also  how  to  place  the  risers 
connecting  with  the  cylinder  of  a  half-turn  platform  stairs  so  that  the  wreath  may  have  a  common  incli- 
nation, making  a  much  superior-shaped  rail  and  saving  several  inches  more  room  than  by  the  method 
i^iven  at  FiG.  2.     See  FiGS.  3.  4  and  5. 

PLATE  24. 

Fixing  the  position  of  risers  connecting  with  15"  cylinders  at  the  starting  and  at  the  landing  of 
straight  fliglits  of  stairs,  so  that  the  over-wood  may  be  removed  in  the  most  desirable  way  to  shape  the 
wreath-pieces,  and  raise  the  height  required  from  the  floor. — Figs,   i,  2  and  3. 

Plan  of  stairs  with  the  top  riser  placed  at  the  diameter-line  of  a  15"  cylinder;  management  of  hand- 
rail, etc. — Figs.  4,  5  and  6. 


PLATE  25. 

Fig.  1.    Plan  of  the  top  portion  of  a  staircase  winding  one  quarter,  with  a  small  cylinder. 
Fig.  2.    Elevation  of  treads  and  rises;   development  or  unfolding  of  the  centre  line  of  wreath.  Plans 
of  hand-rail  and  face-moulds,  FiGS.  3,  4,  5  and  6. 


PLATE  26. 

Fig.  1.    Improved  plan  of  the  top  portion  of  a  staircase  winding  one  quarter,  with  a  small  cylinder. 
Fig.  2.    Elevation  of  treads  and  rises ;   development  of  the  centre  line  of  wreath ;   length   of  balusters. 
Plans  of  hand-rail  and  face-moulds.  Figs.  3,  4,  5  and  6. 


PLATE  27. 

Fig.  1.    Plan  of  the  bottom  portion  of  a  staircase  winding  one  quarter,  with  a  10"  cylinder. 
Fig.  2.    Elevation  of  treads  and  rises  from  plan  FiG.  i ;   also  development  of  the  centre  line  of  wreath, 
giving  lengths  of  balusters  under  wreath.     Plans  of  hand-rail  and  face-moulds,  FlGS.  3,  4,  5  and  6. 

PLATE  28. 

Fig.  1.    Plan  of  the  top  portion  of  a  winding  staircase,  making  a  half-turn,  with  a  10"  cylinder. 
Fig.  2.    Elevation  of  treads  and  rises  and  development  of  the  centre  line  of  wreath,  giving  lengths  01 
balusters,  etc. 

Figs.  3  and  4.  Plan  of  hand  rail  and  face  mould.  Fig.  4.  Face-mould  for  both  quarters  of  the  cylinder, 
the  easement  at  the  top  finishing  to  a  level  from  the  upper  end  of  wreath. 


CONTENTS. 


XV 


PLATE  29. 

Plan  of  the  bottom  portion  of  a  winding  staircase  maiiing  a  half-turn,  with  a  lo"  cylinder. 
Fig.  1.    Elevation  of  treads  and  rises  as  given  at  plan  FiG.  i  ;   also  unfolding  the  centre  line  of  wreath, 
giving  lengths  of  balusters,  etc. 

Fig.  2.    Plan  of  hand-rail  and  face-mould. 
Figs.  3  and  4.    Face-mould  for  both  quarters. 

PLATES  30,  31,  33. 

At  the  starting  of  a  first  flight  of  stairs  the  front-string  is  frequently  curved  out,  the  curve  extending 
from  one  to  five  treads. 

Management  of  Strings  and  Hand-rails  of  Curve-outs. 

Plate  30. — Fig.  i.  Plan  and  elevation.  FiG.  i.  Parallel  pattern.  FiG.  3.  Plan  and  elevation  of  the  starting 
of  a  staircase,  with  the  front-string  curved  out  and  the  newel  set  on  top  of  the  first  step.  FiG.  4.  Parallel 
pattern  for  curve-out  of  hand-rail,  easing  to  a  level  at  newel. 

Plate  31. — Fig.  i.  Plan  and  elevation  of  the  starting  of  a  staircase,  with  the  front-string  curved  out  and 
the  newel  set  on  top  of  the  first  step.  This  plan  is  the  same  as  that  of  FiG.  3,  Plate  30,  but  in  this 
case  the  hand-rail  is  brought  straight  down  to  the  newel,  not  easing  to  a  level  as  at  Plate  30.  FiG.  2. 
Face-mould  for  FiG.  i.  FiG.  3.  Plan  of  a  quarter-platform  stairs  with  a  quarter-cylinder  of  8"  radius,  the 
risers  improperly  set  at  the  chord-lines,  thereby  making  it  necessary  to  get  out  the  quarter-wreath  in  two 
pieces.     Fig.  4.   Parallel  pattern,  two  of  which  make  the  quarter-wreath. 

Plate  32. — Fig.  i.  Plan  of  curve-out, — with  four  steps, — its  shape  designed  to  make  a  proper  connection 
with  a  square  newel,  where  the  sides  of  the  newel  are  required  to  set  parallel  to  the  side-walls  of  the  hall- 
way. Fig.  2.  Elevation  of  treads  and  rises  from  plan  Fig.  i.  Fig.  3.  Face-mould  from  Fig.  i.  showing 
the  squaring  of  the  wreath-piece  at  the  joints.  Fig.  4.  Plan  of  starting  a  staircase,  with  the  front-string 
curved  out,  embracing  four  treads  of  equal  vfidths  at  the  wall-string  and  front-string.  FiG.  5.  Elevation 
of  treads  and  rises  from  plan  FiG.  4.  FiG.  6.  Face-mould  from  plan  FiG.  4,  showing  the  squaring  of 
wreath-piece  at  the  joints, 

PLATE  33. 

Fig.  1.  Plan  of  the  landing  of  a  straight  flight  of  stairs,  with  the  top  riser  set  in  the  whole  deptl- 
of  a  ten-inch  cylinder,  thereby  saving  five  inches. 

Fig.  2.    Elevation  of  tread  and  rises  from  plan  FiG.  i. 

Fig.  3.    Face-mould  from   plan  FiG.   i  ;   also  showing  the  squaring  of  the  wreath-piece   at   the  joints. 
Fig.  4.    Plan  of  the  starting  portion  of    the    same   flight   of   stairs   given  at    FiG.   i.  with    a  ten-inch 
cylinder,  and  the  starting  riser  set  in  the  whole   radius,  or  depth,  of  cylinder,  saving  another  five  inches. 
Fig.  5.    Elevation  of  tread  and  rises  from  plan  FiG.  4. 

Fig.  6.    Elevation  of  tread  and  rises  same  as  Fig.  5,  showing  the  unfolding  of  the  centre  line  of  wreath. 

PLATE  34. 

Hand-rail  wreath  in  one  piece  over  a  7"  cylinder,  the  landing  riser  set  in  the  radius— 3^"— 

OF   THE  cylinder. 

PLATE  35. 

Hand-rail  wreath  in  one  piece  over  a  7"  cylinder  by  a  different  method  from  that  given 
at  Plate  34.  In  this  case  the  landing  riser  is  placed  at  the  chord-line  of  the  cylinder.  Figs. 
4  and  5.  Wainscoting  for  stairs,  its  heights  and  construction. 

PLATE  36. 

Plan  of  winding  stairs  turning  one  quarter,  with  a  quarter-cylinder  of  8"  radius.  Elevation  from  plan. 
Unfolding  the  centre  line  of  wreath.  Length  of  balusters  under  wreath.  Parallel  pattern  for  wreath-piece. 
See  Plate  5,  Fig.  ii. 

PLATE  37. 

Plan  of  half-turn  platform  stairs.  Where  to  place  the  risers  connecting  with  the  cylinder  when  the 
over-wood  is  to  be  removed  from  the  wreath-pieces  as  shown.  Drawing  the  face-mould  for  the  quarter- 
cylinders.  Plan  of  quarter-turn  platform  stairs,  with  a  quarter-cylinder  of  6"  radius.  Where  to  place  the 
risers  connecting  with  the  quarter-cylinder.  Elevation  of  plan,  and  unfolding  the  centre  line  of  wreath. 
Drawing  the  face-mould.     See  Plate  5,  FiG.  10;  also  Plate  45,  FiG.  i. 

PLATE  38. 

Plan  of  half-turn  platform  stairs,  with  the  risers  next  to  platforn:  set  in  the  whole  depth  of  the  cyl 
inder.  Elevation  from  plan  and  unfolding  the  centre  line  of  wreath.  Lengths  of  balusters  under  wreath. 
Drawing  the  face-mould. 


xvi 


CONTENTS. 


PLATE  39. 

Plan  of  half-turn  platform  stairs,  with  one  riser  set  at  the  centre  of  the  cylinder  dividing  the  half- 
turn  space  in  two  platforms;  the  other  two  connecting  risers  each  set  in  the  cylinder  3".  Elevation  from 
plan.    Unfolding  the  centre  line  of  wreath.    Drawing  the  face-mould. 

PLATE  40. 

Plan  of  half-turn  platform  stairs,  with  two  risers  each  set  into  the  15"  cylintler  6".  The  bottom  riser 
set  into  the  cylinder  at  the  starting  of  the  (light  as  found  on  trial  from  elevation.  Drawing  face-moulds. 
See  Fig.  i,  Plate  7. 

PLATE  41. 

Plan  of  stairs  with  two  quarter  platforms  and  a  tread  between,  making  a  half-turn.  Elevation  from 
plan.    Unfolding  the  centre  line  of  v/reath.    Drawing  the  face-mould. 

PLATE  43. 

Plan  of  stairs  in  which  two  flights  start  and  land  in  connection  with  a  12"  cylinder,  the  upper  flight 
starting  and  turning  one  quarter  with  winders,  the  lower  flight  straight  with  the  landing  riser  set  4"  into 
the  cylinder.  Elevation  from  plan  and  unfolding  the  centre  line  of  wreath.  Drawing  the  face-moulds. 
See  Plate  6,  Fig.  3.  * 

PLATE  43. 

Plan  of  stairs  making  a  quarter-turn,  with  a  15"  cylinder,  a  quarter  platform,  and  a  parallel  step  set 
in  the  centre  of  the  cylinder  and  at  right  angles  to  its  diameter.  Elevation  from  plan  and  unfolding  the 
centre  line  of  wreath.  Drawing  the  face-moulds.  Showing  that  in  some  cases  greater  width  of  wood  than 
the  width  of  face-mould  is  required  for  the  wreath.  This  shows  also  the  manner  of  linding  the  thickness  required. 
See  Plate  7,  Fig.  9. 

PLATE  44. 

Plan  of  stairs  turning  one  quarter,  with  an  18"  cylinder,  a  quarter- platform,  and  three  steps  with 
their  front  ends  curved  to  the  cylinder,  designed  to  take  the  place  of  wniders.  Elevation  from  plan  and 
unfolding  the  centre  line  of  the  wreath,  by  means  of  which  the  lengths  of  balusters  under  the  wreath  may 
be  found.    Drawing  the  face-moulds.     See  Plate  7,  Fig.  8. 


PLATE  45. 

Pl.ui  of  circular  stairs  winding  around  a  circular  post.  Plan  of  stairs  with  close  string,  showing  how 
to  build  the  cylinder  solid  with  veneered  faces.  Plan  of  quarter-platform  stairs,  showing  how  to  place  the 
risers  in  tlie  quarter-cylinder  another  way  (when  desirable),  different  from  the  one  given  at  Plate  37, 
Fig.  5,  and  Plate  5,  Fig.  10. 

PLATE  46. 

Plan,  turning  one  quarter,  with  a  quarter-platform,  designed  to  avoid  winders  by  curving  the  fron^ 
ends  of  the  parallel  steps  to  the  cylinder.  Elevation  from  plan  and  unfolding  the  centre  line  of  wreath, 
by  means  of  which  the  lengths  of  balusters  under  the  wreath  may  be  found.  Drawing  the  face-moulds. 
See  Plate  7,  Fig.  4. 

PLATE  47. 

Plan  of  half- turn  winding  stairs  with  7"  cylinder.  Elevation  from  plan,  and  unfolding  the  centre  line  of 
wreath,  by  means  of  which  the  lengths  of  balusters  under  the  wreath  may  be  found.  Drawing  the  face  mould. 
See  Plate  6,  Fig.  2. 

PLATE  48. 

Plan  of  a  starting  portion  of  a  flight  winding  one  quarter  from  a  newel,  the  hand-rail  wreath  covering 
nearly  a  semicircle  on  the  plan,  worked  in  one  piece  and  no  ramp  used  in  connection.  Elevation  from  pl.m 
and  unfoldmg  the  centre  line  of  wreath,  by  means  of  which  the  length  of  each  baluster  under  the  wreath  is 
given.    Drawing  a  parallel  pattern  for  the  wreath  piece. 

PLATE  49. 

Plan  of_  the  starting  portion  of  a  flight  winding  more  than  a  quarter  from  a  newel,  with  a  20"  cvlmder 
containing  five  winders.  The  wreath  covering  nearly  a  semicircle  on  the  plan,  worked  in  one  piece  and 
no  ramp  used  in  connection.  Elevation  from  plan  and  unfolding  the  centre  line  of  wreath,  by  means  ol 
which  the  length  of  each  baluster  under  the  wreath  is  given.  Sketch,  showing  the  wreath  as  squared 
up.    Drawing  the  face-mf)uld.    See  Plate  7,  Fig.  3. 


CONTENTS. 


xvii 


PLATE  50. 

Plan  for  hand-rail  with  parallel  patterns  for  wreath-pieces,  the  plan  taken  Irom  Plate  6.  FiG.  9. 
Steamship  stairs.  Elevation  from  plan  and  unfolding  the  centre  line  of  wreath,  showing  the  length  ot 
each  baluster  under  the  wreath. 


PLATE  51. 

Plan  of  half-turn  platform  stairs  with  six  parallel  steps,  and  platform  placed  in  a  large  cylinder,  the 
steps  at  their  front  ends  curved  to  the  cylinder.  Elevation  from  plan  and  unfolding  the  centre  line  of 
wreath,  showing  the  length  of  each  baluster  under  the  wreath.  The  wreath  to  be  worked  in  three  pieces. 
Drawing  the  face-moulds.    See  Plate  6,  Fig.  8. 


PLATE  52. 

Plan  of  open-newel  steamship  stairs  with  platform  and  wing  flights.  The  newels  at  the  starting  to 
continue  above  the  upper  deck  and  receive  the  level  hand-rail  and  enclosing  balustrade.  Elevation  from 
plan  of  curve  and  unfolding  the  centre  line  of  wreath.    Drawing  the  face-mould. 


PLATE  53. 

Hand-rail  over  a  circular  flight  of  stairs  starting  with  a  newel.  Plan  of  stairs  given  on  PLATE  7, 
Fig.  10.     Drawing  the  face-moulds. 


PLATE  54. 

Plan  for  starting  the  circular  stairs  given  on  Plate  53,  with  a  scroll  step  and   scrolled    hand-rail  in 

place  of  a  newel.    Construction  of  block  for  scroll  step  and  riser.     Scroll  step  complete.     Face-mould  for 

hand-rail  scroll.    How  to  describe  the  scroll  of  a  superior  form  and  in  a  simple  way. 

PLATE  55. 

Hand-rail  over  an  elliptic  flight  of  stairs.  Drawing  the  face-moulds.  See  Plate  7,  FiG.  11,  for  com- 
plete plan  of  the  stairs. 

PLATE  56. 

Instructions  for  sliding  face-moulds,  or  for  placing  them  in  position  to  plumb  the  sides  of  wreath- 
pieces.    Squaring  wreath-pieces. 

PLATE  57. 

Wholesale-store  stairs.  Panel-work  enclosure.  Construction  of  close  front-string,  paneled  and  capped. 
Management  of  newels  and  hand-rail. 

PLATE  58. 

Plan  of  the  landing  portion  of  a  flight  of  stairs  turning  one  quarter,  with  a  quarter-platform  and  square 
corner-pieces  like  small  low-down  newels  set  in  the  angles,  over  which  is  carried  a  continued  hand-rail,  with 
ramp  and  goose-neck  at  the  landing.  Construction  of  close  front-string.  Elevation  showing  paneled  front- 
string,  corner-piece,  balusters,  and  method  of  continuing  the  hand-rail. 

PLATE  59. 

Plan  of  quarter-platform  open-newel  stairs;  top  portion  of  flight.  Elevation  from  plan,  and  details. 
Laying  out  the  angle  newels. 

PLATE  60. 

Plan  of  quarter-platform  stairs  turning  one  quarter.  Turned  angle-newels,  with  hand  rail  ramped  and 
kneed.     Elevation  and  details. 

PLATE  61. 

Design  for  newels,  baluster,  and  close  front-string. 

PLATE  62. 

Plan  of  stairs  turning  one  quarter  and  arranged  to  avoid  winders  by  curving  the  front  end  of  steps, 
making  the  hitter  parallel,  and  securing  a  platform.  Also  by  this  plan  with  the  small  newel  no  twists  are 
required.  Elevation  from  plan,  showing  length  of  newel  and  its  connections  with  string,  rail,  balusters,  etc. 
See  Plate  5,  Fig.  7. 


xviii 


CONTENTS. 


PLATE  C3. 

Design  for  newel,  hand-rail,  and  balusters;  also  design  and  construction  of  close  front-string. 

PLATE  G4. 

Desif^n  for  an  open-moulded  stair-string,  balusters  and  hand-rail.  The  balusters  to  be  screwed  with 
square-headed  lag-screws  to  the  side  of  the  string,  and  the  heads  ot  the  screws  covered  with  turned  or 
carved  rosettes. 

PLATE  65. 

Design  and  construction  of  close  front-string,  newel  and  balusters. 
PLATE  6G. 

Desi"-n  of  a  turned  and  carved  newel,  carved  string,  and  balustrade.  Design  of  spiral-turned  newels 
and  balu"ster.s,  bracketed  string,  ramp,  and  goose-neck  hand-rail.  Plan  and  elevation  of  a  half-turn  platform 
stairs,  the  plan  so  arranged  that  the  newels  will  be  of  equal  heiglits  from  the  platform. 

PLATE  67. 

Ancient  staircase  at  Rouen,  France.    From  the  Monitcur  des  Architectes. 

PLATE  68. 

Interior  view  of  a  flight  of  stairs  turning  one  quarter,  witli  a  platform  at  the  starting  two  rises  up. 
the  platform  ornamentally  enclosed  on  one  side  with  panel-work  to  match  the  hall  wainscot,  and  above 
which  and  across  the  hall  spindle  screen  panels  between  columns. 

PLATE  69. 

Interior  view  of  a  grand  staircase  and  spacious,  suitably-fitted  hall. 

PLATES  TO,  71,  72. 
Designs  for  newels. 

PLATE  73. 
Paneled  soffit  of  a  circular  stairs. 

PLATE  74. 

A  third  method  of  treating  hand-rail  over  a  large  cyliiuler.    Cheap  way  of  treating  the  set-ofTof  a  newel,  etc. 

PLATE  75. 

Another  plan  of  stairs— starting — avoiding  winders.    Two  ways  of  treating  the  hand-rail,  etc. 

PLATE  76. 

The  use  of  side-moulds  in  hand-railing.  To  cut  large  stone  or  wood  square-top  balusters  to  an  exact  length, 
the  tops  to  fit  the  warped  bottom  surface  of  circular  rail,  etc. 

PLATE  77. 

Plan  of  stairs  with  regular  treads  to  chord  line,  6"  cylinder,  quarter  platform,  and  two  rises  above;  the  wreath 
falling  into  the  inclination  of  regular  treads  without  the  intervention  of  a  ramp.    Two  ways  of  treating  the  rail. 

PLATE  78. 

Treatment  of  hand-rail  for  a  quarter  platform  stairs  with  quarter  cylinder,  8"  radius,  both  rises  at  chords. 

PLATE  79. 
Hand-rail  in  one  piece — platform  stairs  7"  cylinder. 


CONTENTS. 

STONE  WORK. 
PLATE  80. 

Circular  slone-work. 

STONE  STAIRS. 

PLATE  81. 
Plan  of  circular  stone-staircase,  etc. 

PLATE  82. 

Management  of  stone  steps  for  circular  stairs  of  Plate  8i,  stone  wainscoting,  etc. 

PLATE  83. 

Unfolding  the  soffit  of  plan  at  8i,  and  apportioning  the  panels,  etc. 
PLATE  84. 

Hand-railing  for  circular  stone-staircase,  from  plan  Plate  8i. 
PLATE  85. 

Squaring  base  and  cap  mouldings  for  wainscot  of  circular  stone-staircase  of  Plates  8i,  82. 

IRON  STAIRS. 

Scientific  and  artistic  treatment  of  iron  stairs,  Plate  86. 
A  large  number  of  plans  and  designs  for  iron  staircases  given  through  Plates  86  to  104. 

PLATES  105,  106. 
Sections  of  hand-rails  of  various  forms  and  full  size. 

PLATES  107,  108,  109. 
Newels  and  balusters. 

Total  number  of  plates,  112. 


STAIRS. 


Primitive  man  had  little  use  for  stairs;  living  in  a  hole  dug  in  the  ground,  or  a  hut  built  of 
the  branches  and  leaves  of  trees,  or  a  log  cabin,  the  utmost  of  his  virants  were  doubtless  supplied 
with  a  rude  ladder. 

"  We  know  little  of  the  staircases  of  the  Greeks  and  Romans,  and  it  is  remarkable  that  Vi- 
truvius  *  makes  no  mention  of  a  staircase  as  an  important  part  of  an  edifice  ;  indeed,  his  silence 
seems  to  lead  to  the  conclusion  that  the  staircases  of  antiquity  were  not  constructed  with  the 
luxury  and  magnificence  to  be  seen  in  more  recent  buildings.  The  best-preserved  ancient  stair- 
cases are  those  constructed  in  the  thickness  of  the  walls  of  the  proraos — vestibules — of  temples 
for  ascending  to  the  roofs.  According  to  Pausaniasf  similar  staircases  existed  in  the  temple  of 
the  Olympian  Jupiter  at  Elis.  They  were  generally  winding  and  spiral.  Sometimes,  as  in  the 
Pantheon  at  Rome,  instead  of  being  circular  on  the  plan  they  are  triangular.  Very  few  ves- 
tiges of  staircases  are  to  be  seen  in  the  ruins  ot  Pompeii,  from  which  it  may  be  inferred 
that  what  there  were  must  have  been  of  wood,  and  moreover  that  few  of  the  houses  were 
more  than  one  story  in  height.  Where  they  exist,  as  in  the  building  at  the  above  place 
called  the  country-house,  and  some  others,  they  are  narrow  and  inconvenient,  with  rises  some- 
times a  foot  in  height.  ...  In  modern  architecture  the  magnificence  of  the  staircase  was  but 
tardily  developed.  The  manners,  too,  and  the  customs  of  domestic  life  for  a  length  of  time 
rendered  unnecessary  more  than  a  staircase  of  very  ordinary  description."  J 

In  England,  all  staircases  preceding  the  latter  part  of  the  reign  of  Charles  the  Second 
—  previous  to  1660  —  were  either  stairs  winding  round  a  post,  or  the  strings  were  framed 
into  square  newels  without  balusters,  but  close-boarded,  sometimes  plastered  between  the  rail 
and  steps.  Fig.  i  is  an  example  built  about  1557  of  the  close-boarding,  pierced  as  the  style 
began,  and  introducing  afterwards  a  variety  of  designs.  Later  the  flat-moulded  baluster  was 
introduced,  as  shown  by  Figs.  2,  3  and  4,  and  the  carved  balustrade,  shown  at  Fig.  5.  As  the 
turning-lathe — a  new  invention  of  that  time— came  into  use,  turned  balusters  and  newels  were 
adopted,  an  example  of  which  is  given  at  Fig.  6. 

At  Fig.  7  is  given  an  example  of  an  ancient  English  tower  staircase  from  Naworth  Castle, 
Cumberland,  England. 

In  other  parts  of  Europe  the  state  of  the  art  in  all  probability  had  not  progressed  beyond 
the  examples  here  given.  With  the  advancement  of  the  arts  and  the  enormous  increase  of 
wealth  many  costly  and  magnificent  staircases  are  now  built  in  private  dwellings  of  the  people 
of  Europe  and  America.  Of  public  buildings,  a  staircase  in  "  Goldsmiths'  Hall,"  London,  Eng- 
land, is  said  to  have  cost  $150,000.  At  Washington,  D.  C,  in  the  new  hall  of  the  House  ot 
Representatives,  is  a  costly  staircase  built  of  stone  with  a  highly  artistic  bronze  balustrade. 
There  are  many  elaborate  and  expensive  staircases  in  the  capitals  of  Europe  that  are  impor- 
tant features  of  their  public  buildings.  In  the  Grand  Castle  of  Chapultepec,  in  Mexico,  there  is  built 
a  peculiar  double  staircase  that  seems  to  have  no  supports,  and  is  said  to  be  the  only  one  of  its  kind 
in  existence.  It  is  built  of  white  marble  and  brass.  It  is  said  that  the  Emperor  Maximilian  expressed 
his  doubts  to  the  architect  of  the  strength  of  this  staircase;  whereupon,  as  a  test,  a  regiment  of  soldiers 
was  marched  up  and  down  the  stairway  ten  abreast,  thus  demonstrating  its  strength.  There  has 
recently — 1893-94 — been  erected  in  this  city  a  palatial  residence  for  Collis  P.  Huntington,  Esq. — 
J.  B.  Post,  architect;  Ellin,  Kitson  Co.,  contractors — in  which  a  magnificent  white  Vermont  marble 
and  Mexican  onyx  staircase  has  been  built.  This  staircase  is  constructed  in  a  semicircle  thirty-four 
feet  in  diameter,  of  three  flights — altogether  ninety-eight  steps,  each  of  which  weighs  about  a  ton  and 
is  eight  feet  long.  The  soffits  of  two  of  these  flights  are  alike  divided  with  four  bands  running 
parallel  to  the  circle  and  fourteen  radiating  or  cross  bands,  all  richly  carved,  that  together  divide  the 
sofiit  into  thirty-nine  panels  ;  these  panels  are  sunk  four  inches  in  depth,  are  moulded,  carved,  and 
with  carved  rosettes  in  the  centre  of  each  panel.  The  hand-rail  is  of  Mexican  onyx,  fourteen  by  six 
and  a  half  inches,  moulded  and  carved.    The  balusters  of  onyx— one  on  each  step— are  six  inches 


*  Vitruvius,  Marcus  Polii,  a  Roman  architect,  15  B.C.  \  Pausanias,  an  ancient  Greek  writer,  170  A.D, 

:|:  Gwilt's  Encyc.  of  Architecture. 


xxii 


STA/RS. 


square,  and  turned.  A  marble  paneled  wainscot  also  ornaments  the  circular  walls  all  the  way  up 
these  stairways  ;  this  wainscoting  has  a  heavy  onyx  moulding  at  the  base  and  raised  centre  panels  of 
onyx  ;  the  whole  is  capped  with  onyx  half  the  width  of  the  hand-rail,  carved  and  moulded  as  before 
described.  All  the  onyx  and  marble  is  highly  polished.  This  staircase  is  said  to  have  cost  $200,000. 
Iron  and  stone  stairs  are  built  on  similar  principles  to  those  of  wooden  ones,  the  difference  being  in 
tlie  treatment  and  changes  required  by  the  materials. 

When  the  first  fashion  of  continued  hand-rail  called  for  the  skill  of  the  workman, 
the  method  adopted  was  that  of  bending  and  gluing  together  a  number 
of  thin  wood  veneers  about  a  convex  cylinder  built  for  the  purpose.  The 
width  of  these  veneers  equaled  the  thickness  of  the  rail,  and  the  thickness 
of  the  veneers  altogether  made  up  the  width  of  the  rail,  as  shown  by  the 
accompanying  sketch.  The  solid  wreath  of  hand-rail  formed  in  this  way, 
after  being  allowed  to  dry, — which  a  writer  observes  "  took  three  weeks," 
— was  removed  from  the  cylinder  and  carved  into  the  shape  of  rail 
required.  As  late  as  the  year  1826  instructions  in  the  above  method — 
although  the  author  protested  against  it — were  given  in  a  book  published  in  England  by 
M.  A.  Nicholson,  entitled  "  The  Carpenter  and  Joiner's  Companion." 


FlO.  S. 


xxiv 


DEFINITIONS. 


Stairs. — That  mechanical  structure  in  a  building  by  which  access  from  one  story  to 
another  is  obtained. 

Staircase. — The  whole  structure,  consisting  sometimes  of  a  number  of  connecting  flights  of 
stairs,  and  again  of  one  flight  only. 

Well-hole. — The  opening  required  for  a  complete  staircase. 

Right-hand  and  Left-hand  Stairs. — A  stair  is  called  right-hand  if,  when  a  person  is  going 
up  the  stairs,  the  hand-rail  is  on  the  right;  but  if  in  going  up  a  stair  the  hand-rail  is  on  the  left, 
then  the  stair  is  called  a  left-hand  stair. 

The  Run  of  a  flight  of  stairs  is  the  horizontal  distance  from  the  first  to  the  last  riser  in  the 
flight. 

Tread. — The  horizontal  distance  between  two  risers.  One  of  the  equal  divisions  into  which 
the  run  of  a  flight  is  divided. 

Height  of  Story. — The  vertical  distance  from  the  top  of  one  floor  to  the  top  of  the  floor  of 
the  story  above. 

Head-room. — Height  required  to  clear  the  head  in  ascending  or  descending  stairs. 

Rise. — The  vertical  height  between  two  treads — one  of  the  equal  divisions  into  which  the 
height  of  a  story  is  divided. 

Riser. — The  board  forming  the  vertical  portion  of  the  front  of  a  step  to  which  it  is  glued 
or  otherwise  fastened  at  right  angles. 

Step. — The  horizontal  plank  upon  which  we  tread  in  ascending  or  descending  a  stair. 

Nosing. — The  outer  or  front  edge  of  a  step.  It  usually  projects  beyond  the  face  of  the 
riser  a  distance  equal  to  the  thickness  of  the  step,  and  is  rounded  or  moulded. 

Winders. — Steps  of  a  triangular  form  in  plan  required  in  turning  an  angle  or  going 
round  a  curve. 

Scroll  Step. — A  bottom  step  with  the  front  end  shaped  to  receive  the  balusters  round 
the  scroll  of  the  hand-rail;  also  called  a  Curtail  Step. 

A  Flight. — One  continued  series  of  steps  without  a  landing. 

A  Landing. — A  horizontal  resting-place  at  the  top  of  any  flight. 

A  Half-turn  Platform. — A  landing  extending  across  the  well-hole,  embracing  the  widths 
of  two  adjoining  parallel  flights  as  they  land  on  and  start  from  the  platform. 

A  Quarter  Platform. — A  square  landing,  the  sides  of  which  each  equal  the  width  of  its 
connecting  flight. 

Wall-string. — The  string  secured  against  the  wall.  A  plank  prepared  by  mortises  sunk 
into  its  face  to  receive  and  house  the  ends  of  steps  and  risers  on  the  wall  side. 

Front-string. — The  string  on  that  side  of  the  stairs  over  which  the  hand-rail  hangs;  a 
plank  prepared  and  sawed  out  to  receive  and  support  the  front  ends  of  steps  and  risers. 

Open-string. — Same  as  front-string. 

Close  Front-string. — A  front-string  into  which  the  ends  of  steps  are  housed  the  same 
as  a  wall-string,  the  upper  edge  of  the  string  capped  to  receive  balusters  ;  and  its  outer 
face  paneled  or  otherwise  ornamentally  finished. 

Pitch. — Angle  of  inclination. 

Pitch-board. — A  piece  of  thin  board  in  the  form  of  a  right-angled  triangle,  one  of  the 
sides  of  the  right  angle  equal  to  a  rise,  the  other  side  of  the  right  angle  equal  to  a  tread 
of  a  stair.    The  hypothenuse  is  the  pitch  or  angle  of  incluiation  of  the  stairs. 

Cylinder. — A  concave  semicircle,  formed  b}'  gluing  together  hollowed  wooden  staves,  or 
by  bending  over  a  convex  cylinder. 

Quarter-cylinder. — A  concave  quarter-circle  hollowed  out  of  a  solid  piece  of  wood,  or 
formed  by  being  bent  over  a  convex  quarter-cylinder. 

Splice. — Half  the  thickness  of  wood  cut  away  for  a  few  inches  of  its  length,  so  that  it 
can  be  joined  to  another  portion  similarly  treated. 

Facia. — A  casing,  finishing  the  face  of  beams — called  headers — along  the  floor  levels. 

Fillet. — A  band  i^"  wide  by  \"  thick,  nailed  to  the  face  of  a  front  string  below  the  cove 
or  scotia,  and  extending  the  width  of  a  tread  ;  and  also  a  similar  band  mitred  with  the  riser. 
The  fillet  is  also  nailed  to  the  facia  under  the  scotia  along  the  levels. 

Curve-out. — A  concave  curve  of  the  face  of  a  front-string  at  its  starting. 

Board. — Merchantable  sawed  lumber  of  various  widths  and   lengths,  and  one  inch  thick  or  less. 


xxvi 


DEFINITIONS. 


Plank. — Merchantable  sawed  lumber  of  different  lengths  and  widths,  and  more  than  one  inch 
thick. 

Timber  or  Beam. — Sawed  lumber  of  large  size. 

Carriage-timbers. — Permanent  timber  supports  nailed  under  stairs  parallel  to  the  lower 
edge  of  strings 

Newel. — An  ornamental  post  or  column — built  or  turned  solid — of  various  sizes  and 
designs,  set  at  the  foot  of  stairs  to  receive  and  secure  the  hand-rail.  In  open-newel  stairs, 
posts  of  a  small  size  set  at  the  angles  and  into  which  the  strings  are  framed. 

A  Straight  Flight  of  Stairs — Is  one  in  which  all  the  steps  are  parallel  and  at  right 
angles  to  the  strings. 

Quarter-turn  Winding  Stairs. — A  stairs  in  which  the  winders  make  a  turn  of  one  quarter 

to  a  landing  or  to  a  continuation  of  the  flight. 

Half-turn  Winding  Stairs. — A  stairs  in  which  the  winders  make  a  turn  of  one  half  to  a 
landing,  or  to  a  continuation  of  the  flight. 

Circular  Stairs — Are  stairs  with  steps  planned  in  a  circle,  toward  the  centre  of  which 
they  all  converge,  and  consequently  are  all  winders.  These  stairs  may  have  a  circular,  a 
square,  or  octagonal-shaped  wall  ;  and  on  the  front  an  open  cylinder  and  continued  hand- 
rail, or  a  solid  circular  post  with  the  front  ends  of  steps  and  risers  housed  into  the  post. 

Elliptic  Stairs. — Stairs  that  are  elliptic  on  the  plan.  The  treads  are  spaced  on  the  front 
and  wall  strings,  but  being  less  in  width  on  the  front-string,  they  all  converge,  but  not  to 
one  centre  like  those  of  a  circuiar  stairs. 

Newel  Stairs. — Stairs  in  which  newels  are  substituted  for  cylinders  and  continued  hand- 
rail. 

Open-newel  Stairs — Are  so  called  because  they  have  small  newels  arranged  at  the 
angles  of  an  opening  in  place  of  cylinders.  The  connecting  front-strings  are  framed  into  the 
newels. 

Hand-rail.— A  variously-moulded  form  and  size  of  rail  running  parallel  to  the  inclination 
of  a  stairs,  and  usually  kept  at  a  vertical  height  of  2'.2"  from  the  top  of  step  to  the  bottom 
of  rail,  at  the  centre  of  short  balusters.  The  rail  also  continues  around  cylinders,  or  connect- 
ing with  newels  parallel  to  floor  levels,  at  a  height  of  2'.6"  *  from  floor  to  bottom  of  rail. 

Baluster. — A  small  column  made  of  different  forms  and  sizes,  but  commonly  turned 
They  are  set  vertically  on  the  steps  of  stairs,  generally  two  on  a  step,  and  placed  the  same 
distances  apart  on  the  floor  levels,  forming  an  ornamental  enclosure  and  furnishing  support 
for  the  hand-rail. 

Balustrade. — Balusters  and  hand-rai'i  combined. 

Wreath. — The  whole  of   a    helically-curved    hand-rail,  whether   it  makes    a  half-revolution 

or  more. 

Wreath-piece. — A  portion  of  a  wreath  less  than  the  whole. 

Face-mould. — A  section  produced  on  any  inclined  plane  vertically  over  a  curved  plan  of 
hand-rail  ;  used  on  the  face  of  plank  to  mark  the  shape  of  the  sides  of  wreath-pieces. 

Side-mould.— A  pattern  of  the  exact  shape  of  the  top  and  bottom  of  wreath,  unfolded  from  the 
centre  line  of  rail-plan — made  of  thin  sheet-zinc  or  straw-board — to  be  applied  to  the  concave  and 
convex  sides  of  wreath-pieces.    See  Plates  20,  21  and  76. 

Ramp. — A  concave  or  convex  curve  or  easement  of  an  angle,  as  sometimes  required  at 
the  end  of  a  wreath,  and  the  adjoining  straight  rail,  where  the  two  have  different  inclinations 

Ramp  and  Knee. — A  concave  easement  of  hand-rail  with  its  upper  end  forming  an  angular 
knee.  When  the  knee  is  curved  convex  the  combined  curves  are  called  a  Swan-neck.  These 
two  forms — rainp  and  knee,  and  ramp  and  swan-neck — are  used  in  open-newel  and  other 
stairs  where  the  newels  are  turned. 

Butt-joint. — An  end  joint  made  at  right  angles  to  the  central  tangent  of  a  wreath-piece; 
and  also  an  end  joint  made  at  right  angles  to  any  straight  length  of  hand-rail. 


*  In  figuring  measurements  of  architectural  drawings,  and  in  specifying  sizes,  feet  and  inches  are  designated  by  accent- 
marks— called  indices — as  follows  :  2'.  6"  meaning  two  feet  six  inches.  Feet  are  denoted  by  one  accent-mark  over 
the  number,  and  a  period  on  the  right  separating  it  from  the  fractions  of  a  foot, — inches  Inches  have  two  accent- 
marks  over  the  number  as  shown.  Feet  and  no  inches  are  indicated  thus:  25'. o" — twenty-five  feet  no  inches  ;  inches 
and  no  feet  thus:  o'.6" — no  feet  six  inches.    The  latter  is  frequently  written  with  the  indices  for  inches  only,  as  6". 


BOOKS  PUBLISHED  TREATING  ON  STAIR-BUILDING. 


The  following  is  a  complete  list  and  dates,  as  far  as  ascertained,  of  publications  in  the  English 
language  that  either  treat  i)artially  or  wholly  of  stair-building  and  hand-railing  : 

Date. 

1693.  Moxon,  "  Mechanical  Exercises." 

1725.  Halfpenny,  "  Art  of  Sound  Building." 

1735.  Francis  Price,  "  British  Carpenter." 

1738.  Batty  Langley,  "  Builder's  Complete  Assistant." 

1741.  "  Workmen's  Treasury." 

1741.  "  Builder's  Jewel." 

1750.  Abraham  Swan,  "  The  British  Architect;  or.  The  Builder's  Treasury  of  Staircases." 
1792.  Peter  Nicholson,  "Carpenter's  Guide." 
1813.  Peter  Nicholson,  New  "Carpenter's  Guide." 

1826.  M.  A.  Nicholson,  "  Carpenter,  Joiner,  and  Builder's  Companion." 
1864.  Joshua  Jeays,  "  Orthogonal  System  of  Hand-railing." 
1873.  Newland's  "  Carpenter  and  Joiner's  Assistant." 

The  above  are  all  English  publications.  The  following  are  all — or  have  been— published  in  the 
United  States  : 

Date. 

1838.  Minard  Lafever,  "  Staircase  and  Hand-rail  Construction." 

1844.  R.  G.  Hatfield,  "  The  American  House-carpenter." 

1845.  Simon  De  Graff,  "  The  Modern  Geometrical  Stair-builder's  Guide." 
1849.  Cupper's  "  Hand-railing." 

1856.  Robert  Riddell's  "Hand-railing." 

1858.  Perry's  "  Hand-railing." 

1859.  Esterbrook  &  Monckton's  "American  Stair-builder." 

1872.  Monckton's  "National  Stair-builder." 

1873.  Wm.  Forbes's  "  The  Sectorian  System  of  "  Hand-railing." 
1873.  Louth's  "  Hand-railing." 

1875.  Gould's  "  Hand  railing." 

1886.  Frank  O.  Cresswell's  "  Hand-railing  and  Staircasing." 
1889.  John  V.  H.  Secor,  "  Nonpareil  System  of  Hand-railing." 

1888-94.  Monckton's  "  Stair-building:  Wood,  Iron,  Stone,  and  New  One-Plane  Method  of  Drawing 
Face-Moulds;  Unfolding  the  Centre  Line  of  Wreaths  and  Side  Moulds." 


SUGGESTIONS. 


The  Attention  of  Teachers  Engaged  in  Giving  Instruction  in  Architectural  Drawing 
in  Technical  Schools  is  Invited  to  the  Examination  of  this  Work,  which  is  believed  to  be 
well  suited  for  taking  up  an  important  part  of  interior  use  and  decoration— the  planning  and 
construction  of  stairs,  a  branch  of  building  but  slightly  touched,  while  roofs  and  other  parts  of 
building  structures  are  taught  in  much  detail. 

Apprentices  who  desire  to  master  the  contents  of  this  book  are  informed  that  much  care  in 
its  preparation  was  given  to  make  the  whole  so  simple  and  clearly  stated,  that  it  would  be  easy 
to  learn  ;  yet  if  any  have  had  no  previous  practice  in  the  use  of  drawing-instruments,  and  no 
knowledge  of  practical  geometry,  such  should  at  once  begin  tiiat  study.  A  notice  of  a  little  book  of 
Practical  Geometry,  treating  likewise  of  the  use  of  instruments  and  all  the  necessities  of  a  beginner  in  the 
study  of  drawing,  will  be  found  in  the  last  pages  of  this  work. 

In  the  Study  of  Hand-railing  the  paper  formed  solids  beginning  with  Plate  No.  lo  should 
be  drawn  as  directed,  cut  out,  and  glued  in  shape  as  explained;  for  this  purpose  a  little  bottle 
(with  brusli)  of  LePage's  liquid  glue  is  best  and  most  convenient. 

The  Squaring  of  Model  Wreath-pieces,  one  quarter  of  full  size — or  one  half  size  in  case 
of  small  cylinders — out  of  some  soft  wood  brings  valuable  results  to  the  apprentice,  of  which  the 
first  is  Practice;  second,  Experience ;  third,  Knowledge. 

Fitting  Wreaths  over  Circular  or  other  Curved  Iron  Staircases.— Begin  by  chalking  on 
the  iron  hand-rail  plate  suitable  lengths  of  wreath-pieces;  and  to  get  a  parallel  pattern  for  each 
of  these  pieces,  press  a  thick,  strong  sheet  of  paper  to  the  top  of  the  plate,  and  mark  the  concave 
and  convex  edges  of  the  plate  as  far  as  the  length  requires;  parallel  to  the  curves  thus  marked 
set  off  each  side  enough  to  make  the  width  and  thickness  equal  that  given  by  trial  as  at  Plate  No.  23, 
Fig.  5,  or  Figs.  3  and  5,  Plate  No.  43;  cut  the  paper  joints  to  suit  the  eye,  and  a  little  long.  These 
paper  patterns  are  used  to  mark  the  shape  of  the  wreath-pieces  on  the  plank;  they  are  then  sawed 
out  square  through.  The  bottom  of  the  wreath-piece  is  first  fitted  to  the  iron  plate  and  the  plate 
let  in  flush  ;  then  plumb-lines  are  put  on  the  joints,  the  sides  worked  plumb  and  brought  to  a 
width;  lastly  the  top  is  cut  away  to  the  thickness,  and  the  joints  finished;  when  the  adjoining 
pieces  are  squared,  their  joints  are  fitted  against  the  first  piece,  etc. 

Note. — Fitting  wreaths  over  iron  staircases  on  iron  rail  plates  is  done  as  above  directed  because  the 
curves  to  which  the  iron  is  brought  are  often  various  and  eccentric,  consequently  fitting  is  resorted  to  as 
quickest  and  best. 

Self-supporting  Circular  Stairs  are  rarely  built.  These  stairs  stand  disconnected  and  away 
from  walls  or  any  points  of  support,  except  at  the  top  and  bottom,  and  have  hand-rails  and  balusters 
over  both  strings.  No  carriage-timbers  need  be  used  if  the  risers  are  increased  in  thickness  and 
the  strings  are  made  thick  and  bent  laminated  as  explained  at  Plate  No.  8,  Fig.  5.  The  strings 
at  the  bottom  should  be  run  down  between  the  floor-beams  and  secured  with  strong  iron  bolts  ; 
they  should  also  be  strongly  bolted  at  the  top.  Only  screws  ought  to  be  used  on  such  a  stair- 
case. Jib  panels  should  be  built  as  high  as  can  be  permitted  under  the  strings  at  the  foot  of 
the  stairs.     The  management  of  hand-railing  for  circular  stairs  is  given  at  Plates  Nos.  53  and  54. 

Close  Paneled  Strings  are  best  suited  to  neweled  stairs,  but  if  this  construction  is  used 
with  cylinders  they  should  be  of  large  diameter,  otherwise  the  work  will  appear  cramped  and 
ugly. 

The  Joints  of  All  Wreath-pieces  with  the  exception  of  those  given   at  Plates  No.  34 
and  35  must  be  made  at  right  angles  to  the  face  of  the  plank. 
Newel  Caps  Mitred  to  Hand-rails  ought  to  be  abolished. 

Turned  newels  should  be  finished  in  one  solid  piece.  The  connection  of  hand-rail  with  newel 
is  stronger  and  better  if  run  straight  to  the  newel  and  bolted  together. 

Balusters  with  Square  Bases  insure  a  stronger  balustrade  than  those  with  circular  bases. 

Hand-rails  may  be  Finished  and  Varnished  before  being  set  up,  if  reasonable  accuracy 
be  observed  in  the  drawing  and  in  the  work. 

Hand-railing. — Thorough  students  will  give  careful  attention  and  study  to  each  of  the  subjoined 
statements  and  references,  viz.,  through  a  knowledge  and  application  of  one  simple  geometrical  process,  that 
of  finding  a  level  line  co>nmon  to  both  planes,  the  following  five  important  operations  are  correctly  performed: 
ist.  Measurement  on  one  plane  for  drawing  all  Face-moulds.  See  Plates  Nos.  9  to  19.  2d.  All  angles 
found  that  are  required  for  Squaring  Wreath.  See  Plates  Nos.  9  to  19.  3d.  Unfolding  a  Central  Wreath- 
line.  See  Plates  Nos.  20  and  21.  4th.  Unfolding  convex  and  concave  Side-moulds.  See  Plates  Nos.  20, 
21,  and  76.  5th.  Finding  on  the  horizontal  plane  the  subtension,  or  opening,  of  any  angle  of  tangents  on  a 
cutting  plane.    See  Plates  Nos.  9  to  19. 

It  would  be  of  great  advantage  if  architects  would  count  the  height  of  stories  by  rises  or  at  least  give 
special  attention  to  so  important  a  matter  for  their  stairs. 


PLATE  1. 

Plan  and  Elevation  of  a  Straight  Flight  of  Stairs  with  a  Seven-inch  Cylinder; 
ALSO  A  Plan  and  Elevation  of  a  Platform  Stairs  with  a  Six-inch  Cylinder  Land- 
ing with  Four  Rises  Above  the  Platform. 

Fig.  I.  Plan  of  a  Straight  Flight  of  Stairs. — The  plan  is  given  to  show  the  width  of  the  stairs, 
the  size  and  position  of  the  cylinder,  the  number  and  position  of  the  rises  and  treads  ;  also  by  means  of  the 
shaded  lines  to  show  the  opening  of  the  well-hole,  its  width  and  length  ;  its  width  sufficient  to  receive  the 
width  of  stairs,  the  diameter  of  cylinder  and  the  thickness  of  facia  ;  its  length  sufficient  for  head-room. 

Fig.  2.  Elevation  of  the  Straight  Flight  of  Stairs  Shown  by  Plan  at  Fig.  i.— The 
number  of  rises  is  determined  by  dividing  the  height  of  story— taken  from  the  top  of  the  floor  of  the 
lower  story  to  the  top  of  the  floor  in  the  upper  story — into  any  number  of  suitable  parts  ;  in  this  case  the 
height  of  story  is  io'.4",  equal  to  124",  which  divided  by  16  gives  a  quotient  of  7^",  the  height  of  one  rise.  In 
the  above  manner  the  height  of  any  given  story  must  be  taken  and  divided  into  any  number,  more  or  less,  of 
rises.  The  rod  E  F  shows  the  manner  of  taking  the  height  and  the  division  of  rises.  To  obtain  the  tread,* 
first  find  the  horizontal  distance  that  can  be  taken  for  the  run  of  the  stairs  and  landing  room  ;  which  in  this 
case  is  equal  to  A  D,  14'. 5";  of  this  the  landing  room,  B  D,  must  never  be  less  than  the  width  of  the  stairs,  and 
is  always  better  several  inches  more  ;  therefore  take  B  D,  2'. 9",  for  landing,  and  C  B,  5",  for  depth  of  cylinder  ; 
leaving  AC,  ii'.3",  to  be  divided  into  treads.  There  is  always  one  tread  less  than  the  number  of  rises  in  each 
flight  of  stairs,  because  the  floor  itself  becomes  a  step  for  the  top  rise  ;  so  having  sixteen  rises  in  this  flight, 
the  remaining  ii'.3",  equal  to  135",  must  be  divided  into  fifteen  parts,  which  equals  9"  for  each  tread, 
as  shown  at  plan  and  elevation.  The  line  G  H  is  the  lower  edge  of  the  string-plank,  which  plank  is  sawed  out 
to  receive  and  make  a  finish  with  the  risers  and  steps.  The  dotted  lines  parallel  to  G  H  indicate  the  position 
of  the  supporting  carriage-timbers.  Head-room  is  secured  by  constructing  the  well-hole  of  a  sufficient  length 
so  that  the  tallest  person  in  ascending  or  descending  the  stairs  would  not  be  in  danger  of  striking  the 
head.  Head-room  should  not  be  less  than  ^'.o".  It  is  not  necessary  to  draw  an  elevation  of  steps  and  rises  to 
determine  head-room,  for  that  can  be  learned  from  the  plan  at  Fig.  i  ;  for  example,  count  thirteen  rises  from 
the  top  down  at  J  ;  thirteen  rises,  7^ "each,  equal  8'. 4^"  ;  subtract  from  this  the  thickness  of  floor,  depth  of 
beam  and  plaster,  altogether  10^",  and  there  will  remain  7'. 6"  for  head-room, — if  the  length  of  the  well-hole 
does  not  cover  the  step  J,  Fig.  i. 

Fig.  3.  Plan  of  Platform  Stairs. — Platform  stairs  ascend  from  one  story  to  another  in  two  or 
more  flights,  having  platforms  between  for  resting  and  changing  their  direction.  This  plan  has  but  one  plat- 
form, taking  the  whole  width  of  the  hall,  and  has  four  rises  in  the  upper  short  flight  and  thirteen  rises  in  the 
lower  starting  flight.    The  shaded  lines  show  the  framing  of  the  open  well-hole,  including  the  platform. 

Fig.  4.  Elevation  of  Platform  Stairs  from  the  Plan  Fig.  3.—!  J  is  the  height-rod  showing 
the  division  and  number  of  rises. 

The  head-room  and  tread  are  found  as  before  explained  at  Fig.  2.  Some  attention  must  be  given  to 
the  position  of  platform  K,  so  that  the  height  underneath  has  sufficient  head-room  and  clears  the  trim  or  fan- 
light of  doorway,  if  there  be  any  ;  for  the  platform  may  be  one  or  more  rises  higher  if  space  can  be  spared  to 
add  one  or  more  treads  to  the  starting  flight — these  treads  to  be  taken  from  the  short  landing  flight.  At  L 
is  shown  the  starting  of  a  second  flight  from  that  floor. 

A  Rule  to  Find  the  Correct  Proportion  of  Tread  to  Rise. — To  any  given  rise  in  inches  add 
a  sum  that  together  will  equal  twelve,  double  the  sum  added  to  the  given  rise  for  the  tread  in  inches, — as 
follows  :  given  a  5"  rise  and  7  make  12,  then  twice  7  equals  the  required  tread,  14";  or  again,  given  a  7"  rise 
and  5  make  12,  then  twice  5  equals  the  tread,  10",  etc. 


*  The  tread  is  the  distance  between  risers,  as  M  N  Fig.  2,  without  inchiding  the  projecting  nosing  ;  when  the  projection  of  the 
nosing  is  added  the  whole  is  called  the  step.    The  projection  of  the  nosing  is  usually  made  equal  to  the  thickness  of  the  step. 


Plate  No.  1  , 


1  FT. 


Fig.  I. 


PLATE 


2. 


Step  LADDERS  and  Stoop. 

Fig.  I.  Plan  of  Stepladder. — This  plan  shows  the  thickness  of  the  sides,  the  width  of  the  ladder, 
the  treads  and  number  of  rises,  also  an  extra  width  of  tread,  as  at  P  Q,  which  should  always  be  allowed  at 
the  top  step  of  a  ladder. 

With  P  Q,  ^j4"  deducted  from  the  run  P  M,  there  remains  Q  M,  equal  to  s'.6}4",  or  42}^",  to  be  divided 
by  10,  the  quotient  of  which  is  4%",  the  tread.  The  point  of  the  ladder  N  M,  if  desirable,  may  be  cut  off  on 
the  line  N  0,  and  glued  and  nailed  to  the  back  edge  of  the  ladder,  keeping  the  point  V  to  the  floor. 

Fig.  2.  Elevation  of  Stepladder  given  at  Plan  Fig.  i. — The  height  from  floor  to  landing 
above  is  S'.i}4",  which  divided  by  10  gives  a  quotient  of  9^",  the  height  of  each  rise.  The  sides  of  step- 
ladders  having  from  ten  to  fifteen  rises  should  be  from  5"  to  7"  in  width  and  not  less  than  lya"  thick.  One 
way  to  lay  out  the  sides  of  a  stepladder  is  as  follows  :  let  T  R  equal  the  tread  and  R  S  the  rise  ;  connect 
T  S  ;  take  the  distance  T  S  in  the  compasses  and  mark  on  the  edge  of  the  side  of  ladder  from  V  to  A  ten  spaces, 
and  with  a  bevel  (as  at  X  taken  from  T)  lay  out  the  angle  and  thickness  of  steps  as  shown.  Another  way  to 
lay  out  the  sides  of  a  stepladder  is  to  use  a  steel  square  (^as  at  B),  placing  the  square  at  the  edge  of  the  ladder 
to  the  height  of  rise  and  width  of  tread  as  figured  on  the  square  and  as  many  times  as  there  are  to  be  rises  in 
the  ladder.  The  steps  should  be  let  into  the  sides  of  a  ladder  from  3/s"  to  3^"  ;  3/g"  will  be  sufificient  if  the 
sides  are  i"  thick. 

Steps  are  set  into  the  sides  of  a  ladder  (as  at  Z  Y)  when  the  sides  of  ladder  are  9"  or  10"  wide  and  2"  or 
3"  thick,  as  sometimes  built  in  buildings  used  for  wholesale  stores. 

To  make  a  small  movable  stepladder  strong  and  keep  the  steps  from  working  loose  a  tenon  should  be  run 
through  the  sides  at  three  points  (as  at  i,  2,  3)  and  properly  nailed. 

Figs.  3  and  4.  Plan  and  Isometrical  Elevation  of  a  Double  Stepladder. — Where  space 
is  limited  and  only  occasional  communication  between  stories  is  necessary,  this  ladder  will  answer  the 
requirements,  as  it  can  be  constructed  in  the  cheapest  manner  and  put  up  in  mere  closet-room,  as  shown 
by  the  plan  at  Fig.  3  and  its  perspective.  Hand-rails  should  be  put  up  at  both  sides  of  the  laddei-, 
hung  on  iron  brackets  well  secured  in  the  wall.  At  Fig.  4  there  are  shown  fifteen  rises  of  8"  each,  mak- 
ing a  total  heiffht  of  lo'.o";  and  fourteen  treads  of  9"  each,  occupying  a  run  of  s'-3"- 

Fig.  5.  Elevation  of  Stoop  with  Platform. — The  newel  post  and  balusters  have  no  turned  work, 
but  are  cut  on  the  angles  and  chamfered. 

The  strings  where  there  are  so  few  steps  may  be  laid  out  with  a  steel  square,  or  can  be  laid  out  with  a 
pitchboard  in  the  same  manner  as  inside  stairs.  Fig.  6.  A  pitchboard,  to  be  made  of  thin,  well-seasoned 
wood,  N  M  0,  must  be  made  perfectly  square,  M  N  the  tread  and  M  0  the  rise.  The  grain  of  the  wood 
should  always  run  in  the  direction  of  N  0.  The  edge  of  the  string  C  should  be  jointed,  and  a  pencil-gauge 
distance  equal  to  C  D  run  along  from  the  edge  C. 

The  distance  C  D  is  equal  all  together  to  depth  of  timber,  thickness  of  riser,  and  thickness  of  ceiling 
boards  underneath.  Along  the  line  D  the  pitchboard  is  marked  on  the  stuff  as  many  times  as  there  are  to  be 
rises,  and  then  sawed  square  through  on  the  line  of  treads,  and  cut  mitring  on  the  line  of  rises.  The  dotted 
lines  show  the  correct  position  of  the  hand-rail  to  determine  its  exact  length  ;  the  platform  level  rail  is  raised 
4"  above  the  platform,  so  that  when  the  rail  at  the  centre  of  the  short  balusters  along  the  flight  is  raised  the 
usual  height,  2'.2",  the  level  rail  over  the  platform  will  be  2'. 6",  the  usual  height  for  a  level  rail.  At  the 
newel  G  H  is  raised  3>^",  which  added  to  2'. 2" — the  height  at  the  centre  of  the  short  balusters— makes  the 
height  G  K  at  newel  2'.^}^". 


Note. — The  whole  of  the  above  practical  details  and  directions  relating  to  an  ordinary  stoop  apply  equally  to  inside  stairs  where 
similar  work  is  required. 


PLATE  3. 


Plan,  Elevation  and  Details  of  a  Common  Straight  Flight  of  Stairs.— Stair-building 

Generally. 

Fig.  I.  Plan  of  a  Common  Straight  Flight  of  Stairs  ;  showing  the  width  of  the  fligiit,  the 
thickness  of  wall-string,  the  width  of  treads  and  number  of  treads  and  rises,  the  position  of  the  balusters,  the 
cylinder,  the  width  and  place  of  hand-rail  and  size  of  newel-post. 

Figs.  2  and  3.    Methods  of  Forming  Cylinders  and  Splicing  them  to  Strings. 

Fig.  4.  Wall-string  Laid  Out,  showing  easements  of  angles  joining  floor-base  at  starting  and  landing, 
mortises  laid  out  with  wedge-room  for  steps  and  risers  which  are  to  be  let  into  the  string 

Fig.  5.  Front-string  Laid  Out. — A  B  must  be  sufficient  for  depth  of  timber,  thickness  of  plaster 
and  of  riser.  The  dotletl  lines  at  C  show  the  wood  to  be  left  on  the  string  for  cylinder-splice.  G  F  H  D  E 
is  the  cylinder  opened  out  ;  F  G  must  be  the  depth  of  floor-beam  and  thickness  of  plaster  ;  the  line  D  E  G  is 
the  bottom  edge  of  the  cylinder,  and  at  E  the  curve  is  raised  somewhat  above  the  direction  of  the  line  A  D  in 
shaping  the  edge,  so  as  to  prevent  a  baggy  appearance  the  cylinder  would  otherwise  have  when  in  place. 

Fig.  6.  Step  and  Riser  as  Glued  Together. —  The  whole  thickness  of  the  riser  is  let  into  the 
ploughed  groove  of  the  step  -^/s"  ;  J  is  a  glue-block,  of  which  two  or  more  are  glued  and  nailed  in  place  as  shown 
along  the  length  of  step  and  riser.  K  K  are  dovetail  mortises  cut  in  the  end  of  steps  to  which  the  balusters 
are  fitted,  glued  and  secured  in  position  as  shown  at  Fig.  8.  The  step  and  riser  are  backnailed  together  as  at 
R  ;  from  two  to  four  nails  are  driven  in,  depending  on  the  length  of  steps. 

Fig.  7.  Stair-timbering  and  Rough-bracketing. — This  drawing  represents  a  vertical  section 
cut  through  the  middle  of  a  flight — a  plan  of  which  is  given  at  Fig.  i — showing  an  end  view  of  steps  and 
risers,  rough  board  brackets  L  L,  the  middle  timber  M,  and  piaster  N.  Stairs  3'.o"  wide  and  less  are  usually 
provided  with  two  carriage,  or  supporting,  timbers,  one  of  which  is  used  to  strengthen  the  front-string,  this 
string  being  securely  nailed  to  the  timber  ;  the  other  timber,  M,  is  placed  at  the  centre  of  the  stairs  and  rough 
board  brackets,  L  L  L,  fitted  and  nailed  as  shown  at  alternate  sides  of  the  timber.  At  S  the  nail  through  the 
rough-bracket  is  driven  into  the  back  of  the  step. 

Fig.  8.  Side  Elevation  of  the  Starting  Portion  of  Stairs,  a  Plan  of  which  is  Given  at 
Fig.  I. —  I'he  hand-rail  is  brought  straight  to  the  newel  at  Q,  as  being  a  stronger  and  better  connection  in 
many  ways  than  the  old  plan  of  a  loose  turned  cap  and  easement  of  rail  mitred  to  the  cap.  P  is  a  jib  panel 
which  is  usually  made  as  a  finish  to  the  bottom  of  a  first-story  flight,  and  also  to  receive  the  level  rail,  0,  that 
encloses  the  basement-stair  well-hole.  There  is  no  better  or  stronger  method  of  building  wooden  stairs  than 
what  is  here  described  in  detail,  where  each  step  and  riser  are  glued  together  in  the  manner  shown  at  Fig.  6, 
and  housed  and  properly  wedged  with  glue  and  hard-wood  wedges  in  the  wall-string,  as  shown  at  Fig.  4, — 
also  housed  and  wedged  in  the  same  manner  at  the  front,  if  a  close  front-string  is  used.  Carriage-timbers, 
rough-bracketed  as  before  described  at  Fig.  7,  of  a  size  from  2"  by  4"  to  4"  by  10",  and  from  two  to  five 
timbers — never  less  than  two — to  each  flight,  depending  on  the  width  and  extent  of  the  stairs  and  the  weight 
they  are  expected  to  carry. 

For  good  substantial  work  well-seasoned  materials  should  be  used  throughout.  So  important  is  this  con- 
sidered in  the  larger-sized  timber  that  old  second-hand  timber  is  sometimes  sought  for.  Whole  flights  of 
stairs — with  the  exception  of  circular,  elliptic,  or  some  other  peculiar  form — are  most  economically  finished  by 
being  wedged  and  nailed  together,  trimmed  and  raised  to  their  places  in  the  building  complete  ;  the  support- 
ing timbers  are  easily  put  in  position  afterward.  Generally  the  staircase  may  be  put  up  on  the  dry  brown 
wall,  and  if  made  of  hard  wood  the  steps,  risers,  strings  and  newels  may  be  completely  covered  with  cheap 
heavy  brown  paper  and  thin  rough  boards,  to  remain  as  a  protection  until  the  walls  are  finished  with  white 
plaster,  the  doors  hung,  the  mantels  and  grates  set. 

This  covering  when  put  on  may  be  so  arranged  as  to  enable  the  stair-builder  to  easily  remove  some  six 
inches  of  it,  enough  to  allow  the  hand-rail  to  be  put  up  and  finished,  leaving  the  balance  of  the  covering  until 
no  longer  recjuired. 

A  Staircase  of  Any  Form  of  Plan  may  be  Finished  on  the  Under  Side,  Showing  its 
Construction  with  far  more  elegance  and  variety  than  any  surface  plastering  or  close  panelling  com- 
monly done.  In  this  finish  the  wedged  strings  will  have  to  be  cased  to  conceal  the  wedging.  Both  the  wall 
and  front  strings  should  have  greater  thickness  than  usual  ;  the  front-string  should  be  thick  enough  to 
dispense  with  a  front  carriage-timber,  or  such  timber  may  be  used  and  cased.  If  desirable  the  risers  can  be 
made  of  a  thickness  and  strength  that  no  middle  supporting  string  or  cased  carriage-timber  would  be 
recjuired  :  this  would  leave  an  unobstructed  view  of  panelled  steps  and  risers,  or  other  ornamental  finish. 

The  Old  English  Method  of  Stair-building — which  is  occasionally  followed  in  this  city,  and 
commonly  in  some  portions  of  the  United  States — is  to  construct  rough  timber  carriage-ways  sawed  out  for 
step  and  riser,  with  rough  steps  nailed  on,  to  be  used  for  travel  during  the  process  of  building,  and  to  be 
plaster-finished  as  required  on  the  under  side,  at  the  same  time  with  the  walls  of  the  building.  This  carriage- 
way is  then  cased  with  finished  strings,  steps  and  risers  ;  the  wall-string  and  sometimes  the  front-string — 
where  the  latter  is  to  be  close — are  scribed  to  the  grooved  step  and  riser  and  set  in  this  groove  with  a  3/^" 
tongue  ;  the  projecting  step  nosing  is  sawed  to  fit  against  the  face  of  the  scribed  string.  T/iis  last  mct/ioJ  of 
building  the  bodies  of  staircases  is  not  as  good  as  the  more  modern  one  previously  described,  and  is  also  tnuch  more 
costly. 


Plate  No.  4 


PLATE  4. 


Planning  Winding  Stairs — Drawing  Elevation  of  the  Same— Laying  Out  the  Strings. 

Fig.  I.  Plan  of  a  Staircase  Winding  One  Quarter,  Alike  at  the  Top  and  at  the  Bottom, 
with  Cylinders  lO"  in  Diameter. — It  is  important  in  planning  winding  stairs  of  various  forms — and  this 
example  will  serve  for  all — to  make  the  treads  as  nearly  as  possible  of  a  uniform  width  on  an  established  line  of 
travel,  which  is  about  14"  from  the  front-string  as  shown.  It  is  well  to  increase  the  width  of  the  stairs  a  few 
inches  both  at  the  top  and  at  the  bottom,  for  the  more  convenient  passage  of  furniture  at  these  turns.  In 
making  the  plan  of  stairs,  the  first  thing  to  be  determined  is  the  wall-lines  A  BCD,  next  the  width  of  the  body 
of  the  stairs  from  the  wall  E  to  the  front-string  F.  The  width  of  the  hall  for  this  staircase  will  require  to  be 
7'  between  rough  walls — ^' .0"  width  of  stairs,  10"  diameter  of  cylinder,  3'.o  passageway,  and  2"  thickness  of 
plaster.  From  the  walls  to  the  cylinder,  at  both  the  top  and  the  bottom  of  the  stairs,  the  width  is  made  3'. 2". 
The  line  of  travel  is  drawn  in  position  as  before  mentioned  ;  the  starting  or  first  and  top  or  landing  riser  lines 
are  now  drawn  as  required,  taking  care  that  not  less  in  any  case  than  2"  level  of  the  cylinders,  as  at  X  X,  be 
left  at  both  top  and  bottom  so  as  to  make  a  proper  finish  with  the  facias.  Between  these  starting  and  landing- 
risers,  on  the  line  of  travel,  equal  spaces  are  marked  for  the  width  and  number  of  treads  required  ;  next  the 
treads  in  the  cylinders  and  along  the  line  of  the  front-string  are  marked  as  figured  ;  now  the  lines  of  risers 
are  drawn  from  the  points  of  division  at  cylinders  and  front-string,  through  the  points  of  division  first  made 
on  the  line  of  travel  :  and  this  completes  the  plan. 

Fig.  2.  Elevation  of  the  Plan  of  Stairs,  Fig.  I. — This  elevation  explains  itself  in  connection 
with  the  plan  beneath  :  The  important  points  given  by  the  elevation  are  the  head-room  from  the  top  of  the 
third  step,  G,  to  the  plastered  ceiling,  H,  and  the  length  of  the  well-hole  as  limited  and  shown  by  the  line  G  H. 
It  is  not  necessary  to  set  up  an  elevation  of  a  staircase  to  fix  the  head-room.  The  head-room  may  be  deter- 
mined, and  the  length  of  well-hole,  by  finding  how  many  rises  down  from  the  top — after  subtracting  therefrom 
the  depth  of  floor-beam,  including  floor  and  plaster — would  equal  in  height  y'.o",  or  very  nearly  that. 

Fig.  3.  Laying  Out  of  the  Front-string  and  Cylinder.— The  distance  K  J,  on  the  front-string, 
must  equal  all  together  the  depth  of  timber,  thickness  of  plaster  and  thickness  of  riser.  The  cylinders  are 
spliced  to  the  string  on  the  lines  0  P  and  R  S  ;  the  treads,  as  given  in  the  cylinders  and  string,  agree  with  the 
plan  Fig.  i.  At  the  starting  cylinder,  N  X  equals  10",  the  width  of  facia,  which  is  the  depth  of,floor-beam  (9") 
and  thickness  of  plaster  (i").  The  top  cylinder  requires  a  straight  piece  of  board  the  thickness  of  facia 
glued  at  its  upper  end,  of  sufficient  width  and  length  (as  at  T  X  V  U)  to  produce  an  easing  between  the  lower 
edge  of  the  cylinder  and  the  lower  edge  of  the  facia  ;  this  piece  is  glued  to  the  cylinder  on  the  line  T  X,  and 
joins  the  facia  on  the  line  U  V.  The  depth  to  the  lower  end  of  the  cylinders  is  found,  as  shown,  by  describing 
arcs  of  a  radius  equal  to  K  J  from  the  angles  of  tread  and  rise  as  shown  at  Q  0  and  W  T.  It  is  better  to 
join  the  cylinder  as  at  R,  on  the  straight  line  R  K,  even  if  it  has  2"  or  3"  more  depth  at  that  point  than  at  K  J. 
Cylinders  are  sometimes — generally  in  the  best  ivork — laid  out  with  the  straight  string  in  one  flank,  as  here  shoicn, 
and  the  whole  of  that  portion  for  forming  the  cylinders  up  to  the  lines  0  P  and  R  S  is  cut  away  at  the  back,  leaving 
only  a  thin  veneer  at  the  face,  which  is  bent  over  a  convex  cylinder  and  filled  out  with  staves,  as  described  at  Fig.  4, 
Plate  8. 

Fig.  4.  Wall-string.— This  string  is  the  starting  portion,  A  B,  at  the  plan  Fig.  i.  In  preparing  this 
string  to  join  the  floor-base,  V  S  is  the  height  of  base,  then  S  L  is  the  easing  of  the  angle  of  string  to  the 
level  of  the  base  ;  or  the  string  may  be  mitred  to  the  base,  as  at  Y  0,  R  Y  being  equal  to  V  S.  The  height  of 
string  M  N  above  the  angle-step  must  be  alike  from  the  same  step  as  M  N  at  the  next  string,  Fig.  5  ;  also  the 
.strings  connecting  at  these  angles  may  be  brought  to  a  level,  curved  as  shown,  or  left  angular  ;  but  they  fuust 
in  all  cases  be  brought  to  a  level,  so  that  the  base-mouldings  ivill  properly  connect. 

Fig.  5.  Wall-string.— This  is  the  whole  of  that  portion  of  wall-string  marked  B  C  of  the  plan  Fig. 
I.  The  height  of  the  string  at  0  P  must  be  alike  from  the  same  angle-step  as  at  0  P  of  Fig.  8.  This  string 
is  laid  out  in  two  pieces  spliced  together  at  the  centre,  Z  D  ;  it  may  be  laid  out  in  one  piece  by  the  use  of  a 
mean  tread,  and  in  another  way,  each  of  which  methods  will  now  be  given. 

Fig.  6.  Wall-string  same  as  Fig.  5,  Laid  Out  in  One  Piece  by  the  Use  of  a  Mean 
Tread. — A  mean  tread  is  found  by  adding  together  the  ten  treads  as  figured,  and  dividing  their  sum  by  the 
number  of  rises  (9),  as  follows  :  25  4- 1 7 -I- 13 -f  9 -f  9 -f  9  +  9 -I- 13 -|- 17 -f  23^  =  144^ 9=  i6g'V -h  "  mean  tread, 
which  is  nearly  161^".  Along  any  line,  A  B,  beginning  at  A,  apply  the  mean  tread,  A  C,  and  the  rise,  C  D,  as 
shown,  marking  each  tread  in  their  order  of  the  width  required,  and  if  the  work  has  been  correctly  performed 
the  last  tread,  25",  will  come  out  at  B  of  the  line  A  B. 

Fig.  7.  To  Lay  Out  Winder-strings  from  a  Scale  Drawing.— Set  up  to  a  scale  0(1}^"  to  i' 
an  elevation  of  treads  and  rises  the  same  as  at  Fig.  6,  and  draw  a  line  touching  the  outermost  points  of  the 
upper  edge  of  string,  as  X  E  0  ;  then  with  a  bevel  set  to  the  angle  Q  0  S  a  string  may  be  laid  out  full  size 
whose  points  X  and  E  will  touch  the  edge  of  plank.  Begin  laying  out  with  the  line  0  S,  and  make  0  S  as 
many  inches  full  size  as  it  measures  on  the  scale. 

Fig.  8.    Wall-string. — This  string  is  the  landing  portion  of  wall-string  marked  C  D  on  the  plan  FiG.  i. 


PLATE  5. 


Through  the  drawings  given  in  this  plate  and  the  two  following  plates,  over  thirty  different  plans  of  stairs 
are  presented  ;  they  are  all  made  to  a  scale  and  figured  for  convenient  reference.  These  various  plans  are 
intended  to  afford  opportunity  for  the  examination  and  study  of  stair  plans,  properly  arranged  for  their 
different  requirements.  The  grading  of  treads  next  to  the  cylinder  in  the  case  of  winders,  so  that  the  wreath 
will  make  easier  curves  and  less  inclinations  in  its  connections,  is  a  matter  of  no  slight  importance.  A  little 
more  attention,  a  better  knowledge  of  practical  details  in  planning  stairs,  will  often  lead  to  saving  valuable 
space,  or  to  a  more  comfortable  passage  from  floor  to  floor.  A  superior  plan  of  stairs  may  even  prove  to  be 
a  question  of  humanity  ;  a  cruel  thing  it  may  be  to  a  little  child  or  an  aged  and  feeble  person  to  subject  them 
to  the  danger  and  discomfort  of  travelling  over  oblique  winding  steps;  as,  for  example,  at  Fig.  5,  when  a  very 
little  more  space,  as  figured,  will  permit  a  safe  and  easy  stairway  for  all,  as  shown  at  Fig.  6. 

Fig.  I.  Plan  of  a  Straight  Flight  of  Stairs  Starting  and  Landing  with  Small  Cylin- 
ders.—  I'he  i)osition  of  cylinders  with  regard  to  starting  and  landing  risers,  as  shown  in  this  case,  is 
explained  in  detail  at  Plate  No.  22,  Figs,  i  and  2. 

Fig.  2.  Plan  of  a  Straight  Flight  of  Stairs  Starting  with  a  Newel  Set-off  2>^"  and 
Landing  with  a  7"  Cylinder ;  the  Landing  Riser  Set  Into  the  Cylinder  2>^". — The  set  off  of 
a  newel  and  its  management  in  connection  with  the  hand-rail  is  given  at  Plate  31,  Figs,  i  and  2.  The  cylin- 
der at  the  landing  is  treated  in  detail  at  Plate  33. 

Fig.  3.  Plan  of  Winding  Staircase  with  Mortised  Strings  Both  Sides. — These  stairs  are 
only  used  where  room  for  better  cannot  be  spared,  in  such  places  as  an  attic  (^r  basement  story.  A  and  B 
are  plank  continuations  of  string  5"  wide  and  long  enough  when  spliced  to  the  mortised  string  at  the  top  and 
at  the  bottom  to  receive  the  winding  steps  and  risers,  which  will  be  better  understood  by  examining  the 
elevation  of  string  set  up  at  Fk;.  4. 

Fig.  5.  Plan  of  the  Top  or  Landing  Portion  of  a  Quarter-turn  Winding  Stairs,  with  a 
Small  Cylinder. — The  management  of  the  hand-rail  of  this  case  is  given  at  Plate  25. 

Fig.  6.  Plan  of  the  Top  Portion  of  a  Staircase  Turning  One-quarter  to  the  Landing, 
with  Diminished  Steps  Around  the  Cylinder ;  Curved  Risers  and  Platform,— This  plan  is  an 
improvement  on  that  given  at  Fig.  5.  By  curving  the  risers,  winders  are  avoided  and  a  roomy  platform 
secured  with  the  same  small  cylinder.  But  with  the  number  and  width  of  treads  alike,  this  plan  requires  7)^" 
more  room,  as  shown  at  C  D.     llw  hand-rail  of  this  case  is  treated  in  detail  at  Pla'i  e  26. 

Fig  7.  Plan  of  the  Top  Portion  of  a  Staircase  Turning  One  Quarter  with  Diminished 
Steps,  Curved  Risers,  Newel  and  Level  Quarter-cylinder,  and  Platform.— In  this  plan  a  small 
newel  is  introduced  with  a  connecting  level  cjuarter-cylinder  ;  designed  to  take  the  place  of  plan  Fig.  6,  where 
this  is  preferred.  By  this  plan  no  wreath  or  ramp  will  be  required.  A  design  of  nnvel ;  the  plan,  elevation  and 
management  of  this  case  in  detail  ivill  be  found  at  Plate  62. 

Fig.  8.  Plan  for  Starting  or  Landing  of  a  Staircase. — By  using  a  single  newel  and  setting  it 
diagonally,  as  shown,  it  will  be  strong  in  all  of  its  connections.  In  some  styles  of  interior  finish  this  position 
of  newel  would  be  desiral)le. 

Fig.  9.  Plan  of  Stairs  (Combining  Two  Platforms  with  Curved  Risers  Between) 
Making  a  Half-turn. —  The  management  of  the  hand-rail  for  this  plan  of  stairs  is  given  in  detail  at 
Plate  41. 

Fig.  10.    Plan  of  a  Platform  Stairs  Making  a  Quarter-turn  with  a  Quarter-cylinder. — 

Whatever  radius  is  taken  for  the  quarter-cylinder  in  this  description  of  stairs,  in  order  to  make  the  best  form 
of  wreath-piece,  from  E,  the  centre  of  the  hand-rail,  to  risers  F  and  G  must  be  each  half  a  tread.  See  Plate 
37,  Figs.  5,  6  and  7.    Also  Plate  45,  Fig.  5. 

Fig.  II.    Plan  of  Stairs  Turning  One  Quarter  with  Winders  and  a  Quarter-cylinder. — 

In  planning  this  kind  of  staircase  experience  has  proved  that  the  best  shaped  hand-rail  is  produced  by 
bringing  the  rail  at  the  upper  [)ortion,  K,  straight  into  the  wreath-piece  at  the  end  without  a  ramp,  for  this 
reason  :  one  winder  at  K,  above  the  quarter-cylinder,  is  all  that  should  be  allowed.    See  Plate  36. 


Plate  No.  6 


Sca'Le'4  in=Ift. 


PLATE  6. 


Fig.  I.  Plan  of  Platform  Stairs  with  the  Risers  at  the  Platform  Set  Into  the  Cylin- 
der  All  that  can  be  Profitable.— Placing  the  risers  in  the  position  given  on  the  plan  saves  6"  at  both 
the  landing  and  starting  flights  connected  with  the  platform.  The  management  of  the  hand-rail  is  given  at 
Plate  No.  38. 

Fig  2.  Plan  of  a  Winding  Staircase  with  10"  Cylinders,  Making  a  Half-turn  at  Each 
Cylinder. — The  hand-rail  for  this  plan  is  treated  in  full  detail  for  the  top  or  landing  portion  at  Plate  No. 
28,  and  for  the  starting  at  Plate  No.  29. 

Fig.  3.  Plan  of  a  Winding  Staircase,  Two  Flights  Connecting  with  a  12"  Cylinder. — 
The  details  and  treatment  of  hand-rail  are  given  at  Plate  No.  42. 

Fig.  4.  Plan  of  Stairs  with  Newel  Set  Between  Two  Quarter-cylinders. — In  this  case  the 
treatment  of  hand-rail,  if  at  the  top  of  a  flight,  will  be  substantially  the  same  as  that  given  at  P'ig.  2  of 
Plate  No.  24,  and  if  at  the  starting  of  the  flight,  Fig.  i  of  Plate  No.  24. 

Fig.  5.  Plan  of  the  Starting  of  a  Staircase. — Where  the  haU  is  wide  enough  and  it  is  desirable 
to  make  the  flight  broad  and  inviting,  the  front-string  is  curved  out,  embracing  four  or  five  treads.  This  case 
of  hand-rail  is  treated  at  Plate  No.  32,  Figs,  i,  2  and  3. 

Fig,  6.  Plan  of  Platform  Stairs  with  Low-down  Small  Corner  Newels  and  Continued 
Hand-rail. — A  design  and  the  management  of  stairs  and  hand-rail  of  this  plan  are  given  in  detail  at 
Plate  No.  58. 

Fig.  7.  Plan  of  Platform  Newelled  Stairs  with  Wing-flights. — This  staircase,  suitable  for  a 
very  large  hall  of  a  public  building,  is  designed  to  be  wainscoted  and  with  half-newels  at  the  walls,  as  shown, 
running  through  and  ornamentally  finished  at  the  under-side  of  the  stairs.  The  best  effect  given  to  a  stair- 
case of  this  character  is  by  showing  the  whole  open  construction  of  the  under-side,  tastefully  finished,  free  from 
plaster  or  close  soffit  panelling. 

Fig.  8.  Plan  of  Stairs  Making  a  Half-turn,  with  a  Large  Cylinder  Filled  with  Treads 
of  Equal  Width  to  those  of  the  Straight  Portion  of  the  Flights,  and  Curving  the  Ends 
of  the  Risers  so  as  to  Avoid  Winders  and  Secure  an  Ample  Platform.— Full  detail  instruction 
for  the  management  of  the  hand-rail  over  this  plan  is  given  at  Plate  No.  51. 

Fig.  9.  Plan  of  Staircase  Suitable  for  Steamboat  or  Ship,  where  Every  Inch  of  Space 
is  Valuable. — The  requirements  of  a  hand-rail  over  this  plan  are  treated  in  detail  at  Plate  No.  50.  See, 
also,  Plate  No.  52. 


PLATE  7. 


Fig.  I.  Plan  of  Platform  Stairs. — By  placing  the  risers  six  inches  into  both  C3  linders  as  seen  in 
this  plan,  that  amount  of  room  is  saved  in  each  case — a  matter  of  saving  that  is  sometimes  of  much  impor- 
tance.   The  treatment  of  the  liand-rail  over  this  plan  is  given  at  Plate  No.  40. 

Fig.  2.  Plan  of  Double  Platform  Stairs  Made  by  Introducing  a  Riser  at  the  Centre  of  the 
Cylinder. — The  treatment  of  the  hand-rail  over  this  plan  is  given  at  Plate  No.  39. 

Fig.  3.  Plan  of  Winding  Stairs  Making  a  Three-quarter  Turn.— The  management  of  the  hand- 
rail over  the  centre  cylinder  of  this  flight  is  given  at  Plate  No.  47,  and  over  the  starting  portion  at  Plate 
No.  48. 

Fig.  4.  Plan  of  a  Quarter  Platform  Stairs.— By  curving  risers  in  the  manner  here  shown,  a  good 
roomy  square  stepping  plan  is  made  of  what  would  otherwise  be  winders  ;  somewhat  like  those  of  plan  at  Fig. 
6.    The  details  and  management  of  hand-rail  over  this  plan  will  be  found  at  Plate  No.  46. 

Fig.  5.  Plan  of  a  Quarter  Platform  Stairs  with  Newels  Set  in  the  Angles.— The  framing  of 
these  newels  and  their  connections  of  hand-rail  is  given  in  complete  detail  at  Plate  No.  59. 

Fig.  6.  Plan  of  a  Quarter-turn  Winding  Stairs  at  Starting. — The  detailed  instruction  for 
the  management  of  hand-rail  over  this  plan  is  given  at  Plate  No.  27. 

Fig.  7.  Plan  of  the  Top  Portion  of  a  Quarter-turn  Winding  Stairs. — The  management  of 
hand-rail  over  a  plan  similar  to  this  will  be  found  at  Plate  No.  25.  This  plan  shows  a  way  of  curving  the 
risers  so  as  to  save  a  sometimes  much-needed  space  by  lessening  the  distance  from  the  wall  B  to  the  landing 
riser  C. 

Fig.  8.  Plan  of  a  Quarter  Platform  Stairs  Much  the  Same  as  that  Given  at  Fig.  4,  Except 
the  Shape  of  the  Cylinder. — 'I  he  management  of  hand-rail  over  this  plan  is  given  at  Plate  No.  44. 

Fig.  9.  Plan  of  a  Quarter  Platform  Stairs  with  One  Tread  Placed  at  the  Centre  of 
the  Cylinder. — Management  and  detail  of  hand-rail  over  this  plan  will  be  found  at  Pla'j  e  No.  43. 

Fig.  10.  Plan  of  a  Circular  Staircase. — The  dotted  lines  show  the  best  method  of  timbering  a 
staircase  of  this  or  similar  form.  The  practical  treatment  of  hand-rail  over  this  plan  may  be  found  at  Plate 
No.  53  ;  also  at  Plate  No.  54  are  given  full  instructions  for  changing  the  plan  of  the  first  step  to  the  scroll 
form,  the  management  of  that  portion  of  the  hand-rail,  also  the  construction  of  the  scroll  step. 

Fig.  II.  Plan  of  an  Elliptic  Staircase. — This  plan  has  the  treads  on  the  line  of  wall  and  front 
strings  graded  so  that  the  risers  are  placed  in  a  direction  nearly  normal  to  the  curve,  keeping  an  even  tread 
on  the  line  of  travel  ;  which  would  not  be  the  case  if  the  treads  were  made  equal  at  the  wall-string  and 
at  the  front-string.    The  hand-rail  over  this  plan  is  given  in  detail  at  Plate  55. 

Independent  or  Self-supporting  Staircases. — This  kind  of  stairs  derives  no  support  from  wall 
or  partition  ;  they  are  seldom  required,  but  when  called  for  are  mostly  of  a  circular  plan.  An  independent 
straight  staircase  presents  no  difficulty  ;  for  all  that  is  required  of  it  is,  that  it  be  well  secured  at  the  top  and 
bottom,  and  that  the  material  and  construction  have  ample  strength  to  support  the  weight  it  will  be  liable  to 
carry.  Where  the  plan  of  a  self-supporting  staircase  is  circular  with  cylinder  opening  as  at  Fig.  10,*  the 
timbers  at  the  foot  of  the  stairs  R  Q  P  should  be  bolted  to  the  floor-beams,  and  bolted  at  all  their  connections 
up  to  and  including  the  floor-beams  at  the  lajiding.  Jib  panels  should  be  put  in  at  the  starting  of  both 
strings  as  high  up  as  can  be  allowed  ;  or  set  up  a  supporting  column  near  the  centre  of  the  flight.  Or  again, 
if  it  is  convenient,  let  an  iron  bolt  secured  in  an  adjoining  wall  project  sufficient  to  support  the  staircase  at 
about  the  centre  L.  With  the  supports  mentioned  these  stairs  may  be  finished  on  the  under  side  and  made  of 
sufficient  strength  without  timbers  by  the  use  of  thick  laminatedf  strings,  the  steps  and  risers  to  be  well  housed 
into  both  strings.    Iron  screws  only  should  be  used — no  nails. 


*  See  Plate  45. 


t  See  Plate  8,  Fig.  5. 


Fig.  11. 


Scale  '4  in.  =  1  ft. 


Plate  No.  8 


PLATE  8. 


Figs.  I  and  2.  Bending  Wood  by  Saw-kerfing. — This  method  of  bending  is  the  weakest 
practised,  but  owing  to  the  fact  that  it  is  thought  to  be  least  expensive  is  frequently  adopted.  To  find 
the  correct  distance  between  saw-kerfs  for  any  required  radius  of  curvature,  select  a  piece  of  stuff  of 
suitable  length  and  equal  to  the  thickness  of  the  material  to  be  bent,  as  at  Fig.  i.  Let  A  B  equal 
the  thickness  of  stuff,  and  A  C  the  radius  of  the  required  curve  ;  make  a  saw-kerf  at  B  0,  leaving 
a  thin  veneer  A  0  uncut,  nail  the  cut  piece  at  S  K,  and  move  it  from  C  to  D,  or  just  enough  to  close 
the  saw-kerf  at  B  ;  then  C  D  being  the  distance  moved  will  also  be  the  exact  space  between  each  saw- 
kerf.  The  same  gaged  thickness  of  veneer  A  0  must  be  kept,  and  the  same  saw  used  for  the  work  to 
be  done,  as  were  used  in  the  trial  at  Fig.  i. 

Fig.  2.  The  Construction  of  a  Circular  Form  Over  Which  the  Saw-kerfed  Material 
as  Above  Explained  is  Shown,  Bent  in  Position. — E  F  G  is  the  plank  rib  (made  of  three  pieces) 
of  which  two  or  more  are  required,  according  to  the  work  to  be  done.  H  J  L  are  the  staves  which  are 
nailed  to  the  ribs  and  so  complete  the  circular  form.  N  M  is  a  veneer  laid  over  the  form  first,  upon 
which  is  bent  and  glued  the  prepared  saw-kerfed  material  P  Q  R  ;  this  must  be  left  on  the  form  until 
the  glue  is  perfectly  dry.  The  piece  of  saw-kerfed  work  P  Q  R,  should  be  drawn  tight  to  the  veneer 
and  the  form  by  means  of  hand-screws,  as  given  by  one  example,  T  U,  with  curved  blocks,  V  W. 

Fig.  3.  Bending  Wood  and  Keying. — This  form  is  in  plan  the  same  as  Fig.  2,  except  that 
the  rib  £  F  G  is  not  curved  at  its  lower  edge  ; — shaping  the  lower  edge  this  way  is  done  for  the  con- 
venient use  of  hand-screws  in  the  manner  shown  at  Fig.  2. 

By  this  method  of  bending,  the  wood  is  removed  from  the  back  of  the  stuff,  as  at  X  X  X,  etc.,  leaving 
the  thickness  of  a  veneer  at  the  face  ;  then  after  bending,  the  grooves  XXX  are  filled  with  tightly  fitted 
strips  of  wood  (glued  in)  called  keys,  as  at  S  S  S,  etc.  It  greatly  adds  to  the  strength  of  this  bent  keyed 
work  to  glue  on  three  strips  of  veneer, — one  at  each  edge  of  the  keyed  stuff,  and  one  in  the  middle. 
The  glue  should  be  perfectly  dry  before  the  work  is  removed  from  the  form.  The  spaces  between  the 
keys  may  be  determined  by  the  same  method  as  that  used  to  find  the  spaces  between  saw-kerfs. 

Fig.  4.  Bending  a  Veneer  Facing  and  Filling  out  the  Thickness  with  Staves. — The 
wood  is  removed  wholly  from  the  back  of  the  stuff  between  the  points  required,  leaving  a  veneer  facing 
which  is  bent  over  the  form,  and  then  staves,  Z  Z  Z,  etc.,  are  fitted  and  glued  on,  as  shown  in  this  drawing. 

Fig.  5-  Laminated  Work. — Bending  several  thicknesses  of  veneer  together  is  defined  as  lamina- 
ted work.  The  whole  of  the  veneers  required  should  be  heated  and  bent  over  the  form  together  and 
secured  in  place  ;  then  releasing  and  applying  glue  to  one-half,  put  it  back  in  position  again,  and  pro- 
ceed with  the  other  half  in  the  same  way,  pressing  and  binding  solidly  the  whole  together  and  to  the 
surface  of  the  form. 

To  ascertain  what  thickness  of  white  pine  will  bear  bending  without  injuring  its  elasticity,  multiply  the 
radius  of  curvature  in  feet  by  the  decimal  .05  and  the  product  will  be  the  thickness  in  inches: — For  example, 
multiply  a  four  feet  radius  of  curvature  by  the  decitnal  given, — 4'.o"  X  .05=. 20,  equal  to  two-tenths  or  one- 
fifth  of  an  inch  thickness,  that  rvould  bend  without  fracture. 

Fig.  6.  Bending  Stair-strings. — This  drawing  shows  the  construction  of  an  ordinary  quarter 
circle  form  with  the  correct  position  of  a  stair-string  bent  over  it  ;  the  ribs  of  this  form  are  quarter 
circles  and  are  set  parallel  to  each  other  and  at  right  angles  to  the  chord  line  R  P. 

Figs.  7  and  8.  Construction  of  a  Form  for  Bending  Quarter  Circle  Stair-strings,  the 
Ribs  to  be  Set  on  an  Angle  Parallel  to  the  Inclination  of  Such  Strings. — There  are  two 
advantages  claimed  for  forms  built  in  this  way  ;  one  is,  a  saving  of  stuff  ;  the  other,  that  the  form 
occupies  less  room. 

Fig.  8. — Plan  of  a  quarter  turn  of  winders  with  a  circular  wall-string.  D  E,  the  circular  wall- 
string  as  laid  out  from  the  plan  A  B.  At  Fig.  7,  L  M  and  F  H  is  the  position  of  the  ribs  parallel  to 
D  E  the  inclination  of  the  circular  string  ;  F  G  equals  C  A  of  Fig.  8 — less  the  thickness  of  stave — and 
is  the  semi-minor  axis,  and  F  H  becomes  the  semi-major  axis  of  an  ellipse  ;  as  the  shape  of  the  rib 
when  placed  on  the  oblique  line  F  H,  becomes  a  quarter  ellipse.  The  ribs  have  to  be  beveled  on  the 
edge  to  range  with  the  lines  L  F  and  M  H.  as  shown. 

Figs.  9  and  10.  Soffit  Mouldings. — These  mouldings,  placed  at  the  lower  edge  of  stair- 
strings,  have  to  be  carried  around  cylinders,  and  this  work  can  be  done  in  different  ways.  A  cylinder 
may  be  made  of  sufficient  length  and  reinforced — filled  out  by  gluing  on  pieces,  as  at  0  R  S,  Fig.  10 — 
then  the  moulding  is  worked  out  solidly  in  connection  with  the  cylinder.  Another  way  is  to  shape  up 
the  lower  edge  of  the  cylinder  filled  out  as  at  0  R  S,  then  fit  and  shape  two  or  more  solid  pieces — 
depending  on  the  size  of  cylinder — of  a  thickness  and  width  sufficient  to  carry  out  a  moulding  similar 
to  N  P,  Fig.  9. 

Figs.  II  and  12.    To  Find  the  Lengths  of  Cylinder  Staves  that  Include  Winder-treads. 

— Set  up  an  elevation  of  treads  and  risers  sufficient  to  get  the  shape  of  cylinder  in  its  connections  with 
the  straight  string  and  facia,  as  here  shown  and  as  before  fully  explained  at  Fig.  3,  Plate  No.  4.  Divide  the 
opening  out  of  the  cylinder  V  W,  into  three  equal  parts,  V  Z,  Z  X  and  X  W  ;  parallel  to  the  risers  draw 
the  lines  Z  L,  X  S  and  W  T  ;  then  the  length  of  each  stave  and  its  position  is  given  at  M  B,  L  F 
and  S  X.  The  construction  of  this  cylinder  and  the  winder-treads  contained  in  it  are  given  at  Fig.  ii. 
The  manner  of  splicing  and  connecting  a  staved  cylinder  with  a  straight  string  is  given  at  Plate  No.  3, 
Fig.  3. 


PLATE  9. 


The  method  of  one-plane  projection  is  where  the  projection  on  the  horizontal  plane  is  alone  required. 
Merely  illustrative  examples  are  here  i^n-e/i  of  the  practical  application  of  the  one-plane  ?nethoil  in  dra7cnng  face- 
moulds  for  hand-railing. 

Fig.  I.  To  Find  the  Angle  of  Tangents  and  Centre  Line  Over  a  Plan  of  a 
Quarter-circle,  where  the  Tangents  are  required  to  have  a  Common  Inclination. — Let 

A  V  Y  be  the  plan,  with  the  tangents  A  U  and  U  Y  ;  let  A  W  U  and  U  T  Y  be  the  common  angles  of  inchnation  ; 
connect  U  X,  the  level  line  common  to  both  planes  ;  through  Y  and  A  draw  the  line  R  S  indefinitely  ;  on  U  as 
centre  with  U  T  as  radius  describe  the  arc  T  S  and  the  arc  at  R  ;  connect  R  U  and  S  U  ;  bend  a  flexible 
strip  and  mark  a  curve  through  the  points  R  V  S:  then  R  U  S  will  be  the  length  and  angle  of  the  tangents, 
and  R  V  S  the  curve-line  over  the  plan  A  V  Y.  To  find  the  angle  ivith  luhich  to  square  the  wreath-piece  at  both 
joints  : — On  U  as  centre  describe  the  arc  Z  B  ;  connect  B  A  ;  then  the  bevel  at  B  will  contain  the  angle 
sought. 

Fig.  2.  To  Find  the  Angle  of  Tangents  and  the  Centre  Line  Over  a  Plan  of  a  Quarter- 
circle  when  the  Tangents  have  Different  Inclinations. — Let  the  plan  be  A  F  M,  with  the  tangents 
A  B  and  B  M  ;  let  the  two  inclinations  be  A  G  B  and  B  C  M.  To  find  the  level  line  common  to  both  planes  :  Make 
C  Q  equal  to  B  G  ;  parallel  to  M  B  draw  Q  0  ;  parallel  to  M  C  draw  0  N;  connect  N  K  :  then  N  K  is  the  line 
sought.  Parallel  to  N  K  draw  B  I  ;  at  right  angles  to  N  K  draw  M  D  and  A  E;  on  B  as  centre  with  B  C  as 
radius  describe  the  arc  C  D  ;  again,  on  B  as  centre  with  A  G  as  radius  describe  an  arc  at  E  ;  connect  E  B 
and  B  D  ;  bend  a  flexible  strip  and  mark  a  curve  through  E  F  D  ;  connect  E  D  :  then  E  B  D  will  be  the 
lengths  and  angle  of  tangents,  and  E  F  D  the  curve-line  over  the  plan  A  F  M.  To  find  the  angle  with  which 
to  square  the  wreath-piece  at  joint  D  :  Continue  B  M  to  L ;  make  M  L  equal  Q  P  ;  connect  L  K  :  then  the  bevel 
at  L  contains  the  angle  required.  To  find  the  angle  7vith  xvhich  to  square  the  wreath-piece  at  the  joint  E  :  Draw 
I  J  parallel  to  M  K  ;  make  I  J  equal  B  H  ;  connect  J  A  :  then  the  bevel  at  J  contains  the  angle  sought. 

Fig.  3.  Plan  of  Hand-rail  a  Quarter-circle,  with  the  Tangents  to  the  Centre  Line 
A  F  and  F  D,  the  Tangents  to  have  Common  Angles  of  Inclination. — Let  A  G  F  and  F  E  D  be 
the  angles  of  a  common  inclination  ;  connect  F  X  ;  from  K,  parallel  to  F  X,  draw  K  L  ;  from  J,  parallel  to 
D  E,  draw  J  M  ;  through  A  and  D  draw  AC;  on  F  as  centre  with  F  E  as  radius  describe  the  arc  E  C.  To 
find  the  angle  with  which  to  square  the  wreath-piece  at  both  joints  :  Make  F  H  equal  F  I  ;  connect  H  A  :  then  the 
bevel  at  H  contains  the  angle  sought. 

Fig.  4.  To  Draw  the  Face-mould  from  Plan  Fig.  3. — Make  B  C,  B  C  equal  B  C  of  Fig.  3  ; 
make  B  F  at  right  angles  to  B  C  and  equal  to  B  F  of  Fig.  3  ;  connect  F  C  and  F  C  ;  make  F  M  and  F  M  each 
equal  F  M  of  Fig.  3  ;  through  M  and  M,  parallel  to  B  F,  draw  K  L  and  K  L  ;  make  M  L,  M  K,  at  each  side  of 
the  centre  equal  J  L  and  J  K  of  Fig.  3  ;  make  B  N  0  equal  the  same  at  Fig.  3  ;  through  C  and  C  draw  K  P 
and  K  Q  ;  make  C  P  equal  C  K,  and  C  Q  equal  C  K  ;  let  C  S  equal  straight  wood,  as  required  ;  parallel  to 
C  S  draw  K  T  and  Q  R  ;  parallel  to  M  C  draw  K  U  ;  make  the  joints  at  right  angles  to  the  tangents  ; 
through  Q  L  0  L  P  of  the  convex  and  K  N  K  of  the  concave  trace  the  curved  edges  of  the  face-mould. 

Fig.  5.  Plan  of  Hand-rail  a  Quarter-circle,  with  Tangents  to  the  Centre  Line,  Q  Z  and 
Z  X,  the  Tangents  to  have  Different  Inclinations. — Let  Q  M  Z  and  Z  G  X  be  the  required  inclina- 
tions of  the  tangents.  To  find  a  level  line  common  to  both  planes  :  Make  G  H  equal  Z  M  ;  draw  H  E  parallel 
to  Z  X,  and  E  T  parallel  to  G  X  ;  connect  T  P  :  then  T  P  is  the  line  sought.  Parallel  to  T  P  draw  I  J,  Z  A  and 
C  D  ;  parallel  to  Z  M  draw  R  L  ;  parallel  to  X  G  draw  Y  F  ;  at  right  angles  to  P  T  draw  X  W  and  Q  0  ;  on 
Z  as  centre  with  Z  G  as  radius  describe  the  arc  G  W  ;  again,  on  Z  as  centre  with  Q  M  as  radius  describe  an 
arc  at  0  ;  connect  0  W.  To  find  the  angle  with  ndiich  to  square  the  7vreath-piece  at  joint  Q)  :  Draw  A  B  parallel 
to  Q  Z  ;  make  A  B  equal  Z  N  ;  connect  B  Q  :  then  the  bevel  at  B  contains  the  angle  required.  To  find  the 
angle  with  which  to  square  the  wreath-piece  at  joint  W  :  Prolong  Z  X  to  K  ;  make  X  K  equal  H  2  ;  connect  K  P  : 
then  the  bevel  at  K  will  be  the  angle  sought. 

Fig.  6.  Face-mould  from  Plan  Fig.  5. — Make  0  U  W  equal  the  same  at  Fig.  5  ;  on  U  as  centre 
with  U  Z  of  Fig.  5  as  radius  describe  an  arc  at  M  ;  on  0  as  centre  with  Q  M  of  P'ig.  5  as  radius  intersect 
the  arc  at  M  ;  connect  0  M,  M  W  and  M  U  ;  make  M  E  F  equal  Z  E  F  of  Fig.  5  ;  make  M  L  equal  M  L  of 
Fig.  5  ;  parallel  to  M  U  draw  I  F  J,  3  E  V  and  C  L  D  ;  make  L  D,  L  C  and  M  S  4  equal  R  D,  R  C  and  Z  S  4 
of  Fig.  5  ;  make  E  V,  E  3  and  F  J,  F  I  equal  T  V,  T  3,  Y  J  and  Y  I  of  Fig.  5  ;  through  W  draw  I  P  ;  make 
W  P  equal  I  W  ;  through  0  draw  C  R  ;  make  0  R  equal  0  C  ;  make  0  T  straight  wood,  as  required  ;  parallel 
to  0  T  draw  C  N  and  R  Q  ;  make  the  joints  at  right  angles  to  the  tangent  ;  parallel  to  W  M  draw  I  Z  ; 
through  R  D  S  V  J  P  of  the  convex  and  C  4  3  I  of  the  concave  trace  the  curved  edges  of  the  face-mould. 

Face-moulds,  their  Number  and  Character. 


1.  Plan  :  Quarter-circle — one  tanfjent  inclined,  the  other  horizontal. 

2.  "  "  tanfjents       "  alike. 

3.  "  "  "  "  differently. 

4.  "      Less  than  a  quarter-circle — one  tangent  inclined,  the  other 

horizontal. 

5.  "      More  than  a  quarter-circle — one  tangent  inclined,  the  other 

horizontal. 


6.  Plan  :  Less  than  a  quarter-circle — tangents  inclined  alike. 

7.  "  "  "             "             "                   differently  in- 

clined. 

8.  "  Elliptic  or  eccentric — tangents  inclined  alike. 

9.  "  "  "             "           "  differently. 

10.  "  Greater  than  a  quarter-circle — tangents  inclined  alike. 

11.  "  "  "             ''            "          "  differently. 


It  is  believed  that  the  above  list  of  eleven  face-moulds  comprises  all  that  are  required  in  hand-railing. 
There  are  two  face-moulds,  however,  that  are  given  in  this  work  not  on  the  list,  one  at  Plate  No.  34  and 
another  at  Plate  No.  35,  each  of  which  include  the  whole  cylinder — a  semicircle.  These  may  be  called 
compound  face-moulds,  for  the  first  is  explained  at  Plate  No.  14  and  used  at  Plate  No.  34,  with  a  portion 
of  the  curve  more  than  the  face-mould  proper,  and  worked  with  the  wreath,  thereby  completing  the  semi- 
circle in  one  wreath-piece  ;  the  other  is  double  the  face-mould  given  at  Plate  No.  10,  the  two  used  as 
one  and  worked  as  directed  at  Plate  No.  35,  completing  the  semicircle  in  one  wreath-piece. 


Note. — These  face-moulds  are  all  given  in  the  above  order,  beginning  at  Plate  No.  id. 


Plate  No.  9 


Plate  No. 10 


V         B         E  A  B 

Fig.  5. 


F  I  G.  3 


PLATE  10. 


This  plate  is  the  first  of  ten  prepared  for  the  purpose  of  giving  instruction  in  a  simple  and  practical  way 
in  the  scientific  requirements  of  hand-railing,  based  on  a  few  and  easily-applied  laws  of  geometry. 

The  object  and  use  of  the  solids — rectangular,  acute  and  obtuse  angled  prisms — introduced  in  these  ele- 
mentary plates,  may  be  summed  up  briefly  :  as  a  convenient  and  direct  means  of  imparting  to  workmen  this 
branch  of  geometrical  knowledge  ;  as  demonstratmg  the  nnportance  and  use  of  tangents  as  applied  to  hand-rail- 
ing, for  two  of  the  vertical  sides  of  every  prism  given  are  tangent  to  a  curve  described  on  the  base,  and  tangent 
to  its  trace  on  the  cutting  plane.  The  upper  end  of  each  prism  is  cut  inclined  on  one  or  two  angles  of  inclina- 
tion in  the  same  plane,  and  shows  the  actual  relation  of  the  inclined  or  cutting  plane  to  the  horizontal  plane  or 
base  ;  *  or,  as  maybe  again  stated,  exhibits  in  every  case  the  exact  relation  of  a  plan  as  given  on  the  base  and  a 
section  of  the  plan  traced  vertically  on  the  inclined  plane.  The  cutting  plane,  as  produced  on  one  end  of  these 
solids,  is  in  each  particular  case  the  position  of  the  plane  or  surface  of  plank  out  of  which  the  wreath-piece 
has  to  be  worked.  The  face-mould  and  its  tangents  are  found  on  this  plane  ;  therefore  the  face-mould  when 
applied  to  the  plank  gives  the  shape  of  the  convex  and  concave  sides  of  the  wreath-piece,  which  must  hang 
vertically — or  plumb — over  the  curved  plan  beneath.  The  paper  representations  of  solids  fare  to  be  pre- 
ferred because  they  can  be  more  easily  and  conveniently  made  than  wood  solids  ;  and  in  making  them  they 
afford  instruction  in  detail  that  wood  solids  do  not,  because  in  the  formation  of  solids  with  paper  the  surfaces, 
angles  and  curves  all  have  to  be  found  in  their  proper  relation  on  one  plane,  which  it  will  be  seen  is  the 
practice  and  knowledge  required  for  drawing  face-moulds  correctly. 

Fig.  I.  Represents  a  Solid  Block  or  Prism  Standing  Vertically  on  a  Square  Base, 
A  B  D  C,  the  Upper  End  Cut  on  an  Inclined  Plane,  A  E  F  C,  forming  Oblique  Angles  with  the 
Sides  A  C  and  B  D,  and  at  Right  Angles  to  the  Sides  ABE  and  CDF  —Upon  the  base  is  described 
a  quarter-circle,  B  C,  tangent  to  the  sides  of  the  solid  C  D  and  D  B.  As  E  F  is  parallel  to  the  square  base 
C  A  B  D,  it  is  therefore  a  level  line  on  the  cutting  plane  C  A  E  F  ;  and  as  B  D  at  the  base  and  E  F  on  the  cut- 
ting plane  are  level  lines,  any  measurement  taken  on  level  lines  at  the  base,  as  G  H,  J  K  and  L  M,  and  carried 
vertically  to  the  cutting  plane,  as  G  R,  J  P  and  L  N,  and  then  parallel  to  the  level  line  F  E  set  off  as  at  R  S, 
P  Q  and  N  0,  will  give  trace  points  of  a  curve  on  the  cutting  plane  perpendicularly  over  the  plan  curve  at  the 
base. 

Fig.  2.  Construction  of  a  Paper  Representation  of  a  Solid  with  its  Angles,  Surfaces 
and  Curved  Lines  as  Given  in  Perspective  and  Described  at  Fig.  i.— Let  A  B  C  D  be  the  square 
base  of  the  solid,  and  D  F  C  the  angle  of  inclmation  over  the  base  C  D  and  A  B  ;  make  D  V,  B  W  and  B  E  each 
equal  D  F  and  at  right  angles  to  the  sides  of  the  base  ;  connect  E  A,  B  W,  W  V  and  V  D  ;  continue  D  C  to  T, 
and  B  A  to  U  ;  let  A  U  equal  A  E,  and  C  T  equal  C  F  ;  connect  U  T.  On  A  as  centre  describe  the  quarter- 
circle  B  C,  tangent  to  the  sides  of  the  base  C  D  and  B  D  ;  through  anv  points  on  the  curve  H  K  M,  parallel  to 
the  level  line  B  D,  draw  the  lines  H  G  X,  K  J  X  and  M  L  X  ;  make  C  N  P  R  T  equal  C  X  X  X  F  ;  draw  R  S,  P  Q 
and  N  0  parallel  to  U  T,  and  equal  to  L  M,  J  K  and  G  H  :  then  U  S  Q  0  C  will  be  the  trace  of  a  curve  on  the 
cutting  plane  lying  perpendicularly  over  the  plan  curve  B  H  K  M  C.  With  a  sharp-pointed  instrument  scratch 
the  lines  A  B,  B  D,  D  C  and  C  A  ;  cut  out  the  remainder  of  the  figure  and  touch  the  adjoining  edges  with  a 
little  glue  or  thick  mucilage  and  bring  them  together,  leaving  all  lines  on  the  outside,  so  that  their  connections 
may  be  seen  and  understood. 

Fig.  3.  Plan  of  Hand-rail  a  Quarter-circle,  with  One  Tangent  to  be  Inclined,  the 
Other  Level.  —  B  D  and  D  C  are  the  plan  tangents  to  the  centre  line  B  C  ;  let  D  F  C  be  the  angle  of  inclina- 
tion over  the  plan  tangent  C  D  ;  the  tangent  D  B  to  remain  level  ;  parallel  to  D  B  draw  E  Q  0,  J  MO  and 
R  S  0.  The  bevel  at  F  contains  the  angle  with  which  to  square  the  wreath- piece  at  joint  B  ;  joint  C  is 
squared  from  the  face  of  the  plank. 

Fig.  4.  Face-mould  from  Plan  Fig.  3.— Make  B  F  and  F  C  at  right  angles  ;  let  F  C  equal  F  C  of  FiG. 
3,  and  F  B  equal  D  B  of  Fig.  3;  through  B  draw  V  E  at  right  angles  to  B  F;  make  F  0  0  0  equal  F  0  0  0 
of  Fig.  3;  parallel  to  F  B  draw  01,  O  Y  E,  0  X  W  and  C  G  H;  make  B  V  equal  B  E;  make  F  I  equal  D  U  of 
Fig  3;  make  0  Z  equal  T  S  of  Fig  3;  make  0  Y  and  0  E  equal  P  N  and  P  E  of  Fig.  3,  and  0  X  and  0  W 
equal  L  K  and  L  J  of  Fig.  3,  and  C  G  and  C  H  equal  C  G  and  C  H  of  Fig.  3;  through  the  points  V  I  Z  Y  X  G 
of  the  convex  and  E  W  H  of  the  concave  trace  the  curved  edges  of  the  face-mould. 

Fig.  5.  Parallel  Pattern  for  Round  Rail,  or  to  be  Used  Instead  of  the  Face-mould  for 
Marking  the  Wreath-piece  on  the  Rough  Plank.— The  measurements  are  taken  from  the  plan  at 
Fig.  3,  as  indicated  by  the  corresponding  letters.  Through  the  central  points  C  M  Q  R  B  describe  circles  of 
any  radius  required  ;  touching  these  circles  on  the  convex  and  concave  bend  a  flexible  strip  of  wood  and 
mark  the  curved  edges  of  the  pattern. 

For  ordinary-sized  hand-rail  wreath-pieces  may  be  worked  out  of  stuff  as  thick  as  the  width  of  rail,  and  a 
parallel  pattern  about  ^"  wider  than  the  required  width  of  rail. 


*  It  should  be  understood  that  the  bases  of  all  these  solids  are  cut  square,  or  at  right  angles  to  their  length  ;  also  at  the  upper 
end  two  adjoining  sides  of  any  solid  may  be  cut  on  two  different  angles  of  inclination,  or  a  common  angle  of  inclination  ;  or  one  side 
may  be  cut  at  right  angles  to  its  length  and  the  adjoining  side  at  any  inclined  angle  ;  but  in  every  case  the  opposite  sides  must  be  cut 
parallel.    \  Drawing-paper,  such  as  Whatman's,  is  best  to  make  paper  representations  of  solids— pasteboard  is  too  thick  and  clumsy. 


PLATE  11. 


Fig.  I.  Represents  a  Solid  Block  or  Prism  Standing  Vertically  on  a  Square  Base, 
ZXWY,  the  Upper  End  Cut  on  the  Side  Z  X  on  the  Angle  of  Inclination,  XVZ,  and  on  the 
Side  X  W  on  the  Same  Angle,  K  U  V. — On  the  horizontal  plane  or  base,  Z  Q  W  represents  the  plan  of  a 
quarter-circle  to  which  the  sides  of  the  solid  Z  X  and  X  W  represent  plan  tangents  ;  also  the  lines  Z  V  and  V  U 
represent  the  tangents  on  the  cutting  plane.  The  sides  of  this  solid  being  cut  on  a  common  angle  of  inclina- 
tion, the  heights  from  the  base  X  and  Y  to  X  V  and  YT  are  alike,  and  therefore  a  line  drawn  on  the  cutting 
plane  from  V  to  T  will  be  a  level  line  ;  and  at  the  base  a  line  drawn  from  X  to  Y  will  be  a  level  line  common 
to  both  planes.  At  any  points  on  the  curve  at  the  base  parallel  to  the  level  line  X  Y,  draw  0  H  and  P  N  ; 
parallel  to  X  V  draw  H  R  and  N  L  :  parallel  to  V  T  draw  L  M  and  R  S  ;  make  L  M  equal  N  P,  and  R  S  equal 
HO;  make  V  J  equal  XQ;  then  the  curve  Z  SJ  M  U  will  lie  perpendicularly  over  the  plan-curve  at  the 
base  Z  0  Q  P  W. 

Fig.  2.  Construction  of  a  Paper  Representation  of  a  Solid  with  its  Curved  Lines  and 
Angles  as  Given  in  Perspective  and  Explained  at  Fu;.  i.— Let  Y  W  X  Z  be  the  square  base  of  the 
solid  ;  on  Y  as  centre  describe  the  quarter-circle  W  Q  Z  ;  prolong  X  W  both  ways  to  C  and  V,  Z  X  to  E,  W  Y  to 
3,  and  F,  and  Z  Y  to  B.  Let  XVZ  be  the  angle  of  inclination  over  XZ;  make  X  E,  W  K  and  K  F  each  equal 
XV;  connect  K  E  and  F  E:  then  K  F  E  will  be  the  same  angle  of  inclination  over  W  X  as  X  V  Z  is  over  X  Z; 
make  WD,  DC,  Y  B  and  Y  3  each  equal  XV;  connect  C  B  and  3  Z.  Througli  Z  W  draw  A  A;  on  X  as  centre 
with  ZV  as  radius  describe  the  arcs  A  A;  on  Z  as  centre  with  A  A  for  radius  describe  an  arc  at  U;  on  V  as 
centre  with  VZ  as  radius  intersect  the  arc  at  U;  and  again  on  Z  as  centre  with  Z3  for  radius  describe  an 
arc  at  T;  and  on  V  as  centre  with  XY  for  radius  intersect  the  arc  at  T;  connect  ZT,  T  U,  V  U  and  VT;  at 
right  angles  to  Z  T  draw  T2;  from  K  draw  K  1  at  right  angles  to  EF;  at  any  points  on  the  plan-cnrve 
draw  PN  and  OH  parallel  to  YX;  draw  N  G  parallel  to  W  F,  and  H  R  parallel  to  X  V;  draw  RS  and  LM 
parallel  to  VT;  make  L  M,  VJ  and  RS  equal  HO,  XQ  and  N  P;  through  ZSJ  M  U  trace  a  curve  that  will 
lie  perpendicularly  over  the  plan-curve  VV  P  Q  0  Z. 

With  a  sharp-pointed  instrument  scratch  ihe  lines  Z  V,  Z  X,  X  V\/',  W  Y  and  YZ;  cut  out  the  remainder 
of  the  figure  and  touch  the  adjoining  edges  with  a  little  glue  or  thick  mucilage  and  bring  them  together, 
leaving  all  lines  on  the  outside,  so  that  their  connections  may  be  seen  and  understood. 

Fig.  3.  Plan  of  Hand-rail  a  Quarter-circle,  the  Plan-tangents  Z  X  and  W  X  to  Have  the 
Common  Angle  of  Inclination,  XVZ  and  WFX. —  Through  WZ  draw  A  A  with  X  F  as  radius; 
on  X  as  centre  describe  the  arc  FA  and  A:  then  A  A  will  be  the  distance  on  the  cutting  plane  over 
W  and  Z  ;  and  if  lines  be  drawn  from  A  to  X  and  A  X,  then  A  X  and  A  X  will  be  the  length  and  angle  of  tangents 
on  the  cutting  plane.  From  B,  parallel  to  X  Y,  draw  B  J  ;  from  N,  parallel  to  W  F,  draw  N  M.  To  find  the 
angle  for  squaring  the  wreath-piece  at  both  joints  :  Make  X  E  equal  X  G  ;  connect  E  Z  :  then  the  bevel  at  E 
will  give  the  angle  sought.  By  reference  to  Fig.  2  when  put  together  as  a  solid  it  will  be  seen  that  the  line 
T  2,  which  is  parallel  to  the  joint  required  at  U,  is  on  the  inclination  of  the  cutting  plane — or  face  of  plank — in 
that  direction,  and  with  the  line  I  K — which  is  on  the  vertical  plane — will  be  the  angle  of  a  plumb-line  on  the 
butt  joint  of  such  a  wreath-piece  as  this  centre  line  Z  U  applies  to.  2  T  of  Fig.  2  equals  Z  E  of  Fig.  3  ;  K  I 
equals  X  G  E  of  Fig.  3,  and  the  angle  T  2,  I  K  of  Fig.  2  equals  the  angle  Z  E  X  of  Fig.  3. 

Fig.  4  Face-mould  over  a  Quarter-circle,  the  Tangents  of  a  Common  Inclination,  as 
Given  and  Explained  at  the  Plan  Fig.  3.— On  a  line  FZ  make  K  Z  and  K  F  each  equal  A  K  of  Fig.  3  ; 
make  K  V  at  right  angles  to  Z  F,  and  equal  to  K  X  of  Fig.  3  ;  connect  Z  V  and  FV  ;  make  F  N  and  Z  H  each 
equal  F  M  of  Fig.  3  ;  through  H  and  N  draw  T  D  and  J  B,  at  right  angles  to  F  Z  ;  make  V  S  K  equal  X  S  K  of 
Fig.  3  ;  make  H  T  and  H  D,  and  N  J  and  N  B  each  equal  N  J  and  N  B  of  Fig.  3.  Through  F  draw  B  I  ;  make 
F  I  equal  F  B  ;  through  Z  draw  D  I  ;  made  Z  I  equal  Z  D  ;  through  I  T  S  J  I  on  the  convex,  and  D  K  B  on  the 
concave,  trace  the  curved  edges  of  the  face-mould.  The  joints  Z  and  F  are  made  at  right  angles  to  the 
tangents.    The  slide-line  will  be  explained  further  along. 

Fig.  5.  Parallel  Pattern  for  Round  Rail,  or  to  be  Used  Instead  of  the  Face-mould  (as 
a  Means  of  Saving  Stuff)  for  Marking  the  \Vreath-piece  on  the  Rough  Plank.— On  the  line 
A  A  make  K  A,  K  A  each  equal  K  A  of  Fig.  3  ;  at  right  angles  to  A  A  draw  K  X,  equal  to  K  X  of  Fig.  3  ;  connect 
X  A,  X  A  and  make  the  joints  A,  A  at  right  angles  to  A  X  ;  make  A  N,  A  N  each  equal  F  M  of  Fig.  3  ;  make  N  P, 
N  P  each  at  right  angles  to  A  A,  and  equal  to  N  P  of  Fig.  3  ;  make  X  Q  equal  X  Q  of  Fig.  3  .  describe  circles 
on  the  centres  A  P  Q  P  A  of  any  required  radius  for  width  of  pattern.  J^or  ordinary-sized  hand-rails — such  as 
2"  thick  by  3"  wide,  z)/^"  thick  by  3^"  wide,  2  i^"  thick  by  4"  wide — any  wreath-piece  may  be  worked  out  of  stuff  as 
thick  as  the  width  of  hand-rail,  with  a  parallel  pattern  like  Fig.  5  about  voider  than  the  widtJi  of  the 
hand-rail.    See  Plate  No.  56,  Figs.  6  and  7. 

Fig.  6.  Exhibits  the  two  Solids  Presented— that  of  Fig.  i,  Plate  No.  10,  and  Fig.  i. 
of  This  Plate — brought  together,  the  quarter-circle  of  each  completing  the  plan  of  a  semicircle  on  the 
horizontal  plane  and  showing  the  vertical  trace  of  the  semicircle  on  the  cutting  planes  of  the  two  differently- 
cut  solids. 

Fig.  7.  This  Solid  is  Reproduced  Half  the  Size  of  Fig.  i,  merely  to  be  Used  for  the 
Purpose  of  Showing  the  Correctness  of  A  A  and  the  Angles  A  AX,  as  Described  at  Fig.  3. — 

In  this  figure  V  T,  as  before  explained,  is  a  level  line  on  the  cutting  plane,  and  X  Y  the  position  of  a  level  line 
on  the  horizontal  plane  common  to  both  planes.  Now  if  a  vertical  plane  be  conceived  with  Z  W  as  base,  it 
would  touch  F  and  Z  on  the  cutting  plane,  and  F  Z  on  that  plane  would  be  the  distance  required  in  position 
on  the  horizontal  plane.  Extend  the  base  of  the  vertical  plane  WZ  to  A  A  indefinitely.  On  the  horizontal 
plane,  X  being  vertically  under  V  of  the  cutting  plane,  and  V  I  and  X  K  measuring  alike  on  both  planes,  set  one 
foot  of  the  compasses  on  X,  and  with  Z  V  or  V  F  for  radiire  describe  arcs  at  A  A  ;  then  AX  A  and  their  angles 
on  the  horizontal  plane  will  equal  the  angles  Z  V  F  on  the  cutting  plane. 


Plate  No.  11. 


Plate  No. 12 


PLATE  12. 

Fig.  I.  Represents  a  Solid  Block  or  Prism  Standing  Vertically  on  a  Square  Base 
A  B  C  D,  the  Upper  End  Cut  on  an  Inclined  Plane  Containing  the  Two  Different  Inclina- 
tions A  E  B  and  E  G  F. — Let  A  N  C  represent  a  quarter-circle  to  which  the  sides  of  the  solids  A  B  and  B  C 
are  tangent.  To  find  the  direction  of  a  level  line  on  the  cutting  plane  from  the  point  H  :  make  C  I  equal  D 
H,  connect  H  I  ;  draw  I  J  parallel  to  the  base  C  B  :  then  J  H  will  be  the  level  line  sought.  Draw  J  M  parallel 
to  E  B :  then  M  D  on  the  horizontal  plane  will  be  the  direction  of  a  level  line  common  to  both  planes  ;  again, 
from  the  point  E,  a  level  line  E  0  will  be  found  by  making  D  Q  and  T  0  equal  to  B  E  ;  then  B  T  on  the  hori- 
zontal plane  will  be  the  direction  of  a  level  line  as  before.  These  level  lines  T  B,  0  E,  D  M  and  J  H,  being 
all  of  the  same  length,  so  all  level  lines  drawn  on  the  base  to  which  perpendicular  lines  and  level  lines  on  the 
cutting  plane  are  drawn,  and  equal  measurements  taken  from  the  curve  at  the  base  (as  B  N  and  E  P),  will 
give  the  trace  of  the  curve  on  the  cutting  plane  perpendicularly  over  that  at  the  base.  As  the  sides  of  the 
solid  A  B  and  B  C  at  the  base,  are  tangent  to  the  curve,  so  A  E  and  E  G  are  tangent  to  the  curve  traced  on  the  cut- 
ting plane.  A  level  line  eommon  to  both  planes  may  be  demonstrated  as  follows  :  prolong  the  inclination  G  H  until 
it  meets  the  prolongation  of  the  horizontal  plane  C  D  a/  R  ;  also  continue  the  inclination  G  E  until  it  meets  the  con- 
tinuation of  the  base  C  B  S  ;  connect  S  R  :  the/i  S  R  is  the  intersecting  line  of  the  plane  of  the  two  inclinations  G  E 
and  E  A,  and  the  horizontal  plane  C  R  and  C  S  ;  also  S  R  is  the  position  of  a  level  line  common  to  both  planes. 

Fig.  2,  Construction  of  a  Paper  Representation  of  the  Solid  with  Its  Curved  Lines 
and  Angles  as  Given  in  Perspective  and  Described  in  Fig.  i. — Let  A  B  C  D  be  the  square  base 
of  the  solid  ;  on  D  as  centre  describe  the  quarter-circle  A  U  V  C.  Prolong  B  C  both  ways  to  Z  and  N  ;  D  C 
to  Y  and  I  ;  A  D  to  P  and  A  B  to  M.  Let  B  Z  A  be  the  inclination  of  the  plan  tangent  B  A  ;  make  B  M  and  C  K  each 
equal  B  Z  ;  let  K  I  M  be  the  inclination  of  the  plan  tangent  C  B  ;  let  C  0,  D  P  and  D  Y  each  equal  K  I  ;  make  0  N 
equal  B  Z  ;  connect  N  P  and  Y  A.  To  find  the  direction  of  a  level  line  on  the  horizontal  plane  from  the  point  B, 
make  D  2  equal  B  Z,  draw  2  3  at  right  angles  to  Y  D  ;  and  3  X  parallel  to  Y  D  :  then  X  B  will  be  the  direction  of  the 
line  sought  ;  to  find  the  level  line  from  the  point  D,  make  C  J  equal  D  Y,  and  draw  J  L  parallel  to  C  B  ;  from  L 
draw  L  W  parallel  to  C  I  :  then  D  W  will  be  the  direction  of  a  level  line  common  to  both  planes  from  the 
point  D.  From  A  and  C  at  right  angles  to  the  level  lines  X  B  or  W  D,  draw  A  G  and  C  H  indefinitely  ;  with 
Z  A  as  radius  on  B  as  centre  describe  an  arc  at  G,  and  with  M  I  as  radius  on  B  describe  an  arc  at  H  :  then 
H  G  will  be  the  distance  on  the  cutting  plane  over  C  A  of  the  plan,  and  if  lines  are  drawn  from  H  to  B,  and 
from  G  to  B,  the  lengths  and  angle  of  tangents  on  the  cutting  plane  will  be  given.  With  M  I  as  radius  set 
one  leg  of  the  compasses  on  and  describe  an  arc  at  E,  and  with  H  G  as  radius,  on  A  intersect  the  arc  at  E, 
connect  Z  E,  make  Z  T  equal  M  L  ;  on  A  with  A  Y  as  radius  describe  the  arc  Y  F  ;  with  Z  A  as  radius  on  E 
intersect  the  arc  at  F  ;  connect  E  F  and  F  A  ;  connect  FT;  on  A  describe  the  arc  3  Q  ;  connect  Q  Z  ;  make 
Z  R  equal  B  U,  and  T  S  equal  W  V  :  then  the  curve  ARSE  will  be  the  trace  on  the  cutting  plane  perpen- 
dicularly over  the  curve  at  the  base.  From  F  at  right  angles  to  A  F  draw  the  line  F  4,  and  from  F  at  right 
angles  to  F  E  draw  F  6  ;  from  P  at  right  angles  to  P  N  draw  P  8  ;  from  J  at  right  angles  to  I  M  draw  J  5. 
With  a  sharp-pointed  instrument  scratch  the  lines  A  B  C  D,  and  Z  A  ;  then  with  a  sharp  knife  cut  through  the 
outlines  of  the  figure,  and  touch  the  adjoining  edges  with  a  little  glue  or  thick  mucilage  and  bring  them 
together,  leaving  all  lines  on  the  outside  for  examination. 

Fig.  3.  Plan  of  Hand-Rail  a  Quarter-Circle  in  which  the  Tangents  to  the  Centre 
Line  C  B  and  A  B  Require  Two  Different  Inclinations  as  B  E  C  and  A  S  B. — The  plan  of  rail  in 
every  case  consists  simply  of  the  convex  and  concave  curve  lines  embracing  the  width  of  the  rail,  also  the 
centre  curve  line  and  its  tangents  ;  there  has  then  to  be  added  to  this  plan  certain  lines,  which  in  their  posi- 
tion fix  the  kind  of  face-mould  required,  and  supply  points  of  measurement  from  which  to  draw  the  face- 
mould.  Let  the  inclination  of  the  tangents  B  E  C  and  A  S  B  be  first  fixed,  then  find  the  direction  of  a  level 
line  common  to  both  planes  as  follows  :  Make  B  F  equal  A  S  ;  draw  F  J  parallel  to  B  C  ;  from  J  draw  J  I 
parallel  to  E  B  ;  connect  I  D  :  then  I  D  will  be  the  level  line  sought.  Parallel  to  D  I  draw  L  6,  R  B  and  P  X  ; 
from  8  parallel  to  B  E  draw  8  W  ;  from  V  parallel  to  A  S  draw  V  U.  To  find  the  distance  over  C  A  on  the  cut- 
ting plane  :  from  C  and  from  A,  at  right  angles  to  I  D,  draw  C  H  and  A  G  indefinitely  ;  with  C  E  as  radius 
set  one  foot  of  the  compasses  on  B  and  describe  the  arc  at  H,  and  with  B  S  as  radius  on  B  describe  the  arc 
S  G  ;  connect  G  H  :  then  G  H  will  be  the  distance  sought  ;  and  if  B  H  and  B  G  are  connected  the  lines  will 
contain  the  angle  and  length  of  tangents  on  the  cutting  plane.  To  find  the  angles  with  which  to  square  the 
wreath-piece  :  prolong  B  C  to  Z  ;  make  C  Z  equal  I  K;  connect  Z  D  :  then  the  bevels  at  Z  will  give  the  plumb 
line  to  square  the  wreath-piece  at  the  butt-joint  over  C.  Continue  B  A  to  Q  ;  make  A  Q  equal  A  T  ;  connect 
Q  R  :  then  the  bevel  at  Q  will  give  the  plumb  line  to  square  the  wreath-piece  at  the  butt-joint  over  A. 

Fig.  4.  Face-mould  Over  a  Plan  of  a  Quarter-circle,  the  Tangents  of  Two  Different 
Inclinations  as  Given  at  Fig.  3. — Draw  the  line  C  A  and  make  C  0  and  0  A  equal  H  0  and  0  G  of  Fig.  3. 
On  C  with  the  radius  C  E  of  Fig.  3  describe  an  arc  at  E  ;  on  0  with  the  radius  0  B  of  Fig.  3  describe  an 
intersecting  arc  at  E,  and  on  A  with  the  radius  B  S  of  Fig.  3  intersect  the  arc  at  E  ;  connect  C  E,  A  E  and  0  E; 
make  C,  8,  I  equal  C  W  J  of  Fig.  3  ;  make  E  V  equal  B  U  of  Fig.  3.  Parallel  to  0  E  through  8,  I,  V  draw 
L  6,  M  Y  and  P  X;  make  8,  6,  8  U  I  Y,  I  M,  E  4  N  and  V  X  and  V  P  each  equal  the  corresponding  letters  of 
Fig.  3.  Through  C  draw  L  B  ;  make  C  B  equal  C  L  ;  through  A  draw  P  D  ;  make  A  D  equal  A  P.  Through 
L  M  N  P  on  the  concave,  and  B  6  Y  4  X  D  on  the  convex,  trace  the  curves  of  the  face-mould.  The  joints  A 
and  C  are  made  at  right  angles  to  the  tangents  A  E  and  C  E.  The  slide  line  is  drawn  anywhere  on  the  face- 
mould  at  right  angles  to  the  level  line  0  E. 

Fig.  5.  Parallel  Pattern  for  Round-rail  or  to  be  Used  Instead  of  the  Face-mould — as 
a  Means  of  Saving  Stuff— for  Marking  the  Wreath-piece  on  the  rough  Plank. — Make  H  O 
and  Q  G  each  equal  H  0  and  0  G  of  Fig.  3.  The  tangents  H  B  and  G  B,  and  the  level  line  B  0,  are  the 
same  -as  Fig.  4.  Make  H  I  and  B  V  equal  C  J  and  B  U  of  Fig.  3  ;  draw  I,  5  and  V  2  parallel  to  0  B  ;  make 
V  2,  B  3  and  I,  5  each  equal  the  corresponding  letters  and  figures  at  Fig.  3.  The  joints  are  made  at  right 
angles  to  the  tangents.  Describe  circles  an  the  centres  H  5,  3,  2  G,  of  any  required  radius  for  width  of 
pattern.  Fig.  6. — A  solid  similar  to  Fig.  i  introduced  to  call  attention  to  the  two  sections  that  may  be  cut  in  a 
direction  on  the  inclined  plane,  at  right  angles  to  each  of  the  differently  inclined  sides  or  tangents,  as  A  B  and  B  C  ; 
and  also  cut  down  the  sides  of  the  solid  in  a  direction  at  right  angles  to  each  inclination  of  the  cutting  plane  as  A  D 
and  C  E.  The  inclined  plane  of  these  solids  should  be  understood  as  representing  the  surface  and  position  of  rail 
plank  J  the  lines  A  B  and  B  C  the  direction  of  joints  of  face-moulds  j  and  the  lines  A  D  and  C  E  represent  the  joints 
square  through  the  thickmss  of  plank.  The  angle  B  C  E  tvill  square  the  wreath-piece  at  the  butt-joint  F  G  ;  and 
the  angle  BAD  squares  the  wreath  at  the  joint  H  I.  The  sections  here  given  and  described  are  also  outlined  on  the 
paper  solid  to  be  formed  at  Fig.  2  by  the  lines  F  7,  7.  6  and  P  8,  also  F  4  aiid  J  5. 


PLATE  13. 


Fig.  I.  Represents  a  Solid  Block  or  Prism  Standing  Vertically  on  a  Base  A  B  P  0, 
the  Sides  of  Which  are  Parallel,  and  Have  Two  Obtuse  and  Two  Acute  Angles. — The 

upper  end  of  this  prism  is  cut  on  tlic  inclination  P  M  B,  and  M  N  at  right  angles  to  the  sides,  ai.d  parallel  to 
the  base  P  0.  In  this  solid  B  A,  being  in  the  horizontal  plane,  and  also  terminating  the  inclined  plane,  is  a 
level  line  common  to  both  planes.  On  the  base  describe  the  curve  A  D  J  P  tangent  to  the  sides  of  the  solid 
A  B  and  B  P.  To  find  the  trace  of  this  plan  curve  on  the  cutting  plane  :  parallel  to  A  B  on  the  base  and  at 
pleasure  draw  C  D  and  G  J  ;  at  G  and  C  parallel  to  P  M  draw  G  K  and  C  E  ;  from  E  and  K  parallel  to  B  A 
draw  K  L  and  E  F  ;  make  E  F  equal  C  D,  and  K  L  equal  G  J  ;  through  the  points  A  F  L  M  on  the  inclined 
plane  trace  a  curve  perpendicularly  over  the  plan  curve  A  D  J  P.  As  the  sides  of  the  solid  A  B  and  B  P  at 
the  base  are  tangent  to  the  plan  curve,  so  A  B  and  B  M  are  tangent  to  the  curve  traced  on  the  cutting  plane. 

Fig.  2.  Construction  of  a  Paper  Representation  of  the  Solid  With  its  Curved  Lines 
and  Angles  Given  in  Perspective  and  Described  at  Fig.  i.— Let  A  B  P  0  be  the  form  of  the 
base,  the  opposite  sides  of  which  are  parallel  and  equal.  From  A  at  right  angles  to  B  A  draw  A  X  ;  from  P  at 
right  angles  to  B  P  draw  R  X  ;  on  X  as  centre  describe  the  plan  of  curve  A  D  J  P,  tangent  to  the  sides  of  the 
base  B  A  and  B  P.  Let  P  R  B  be  the  inclination — assumed  or  required — over  the  base  B  P  ;  make  0  N  and 
P  M  at  right  angles  to  0  P  and  each  equal  P  R  ;  connect  N  M  ;  make  0  U  at  right  angles  to  A  0  and  equal 
to  P  R  ;  connect  U  A  ;  parallel  to  B  A  from  any  points  on  the  curve  D  and  J  draw  J  G  and  D  C  ;  parallel  to 
P  R  draw  C  E  and  G  K  ;  on  B  with  B  H  as  radius  describe  the  arc  Q  R  S  indefinitely  ;  on  A  with  A  Q  as 
radius  intersect  the  arc  at  S  ;  connect  SB;  on  A  with  A  U  as  radius  describe  an  arc  at  T  ;  on  S  with  B  A  as 
radius  intersect  the  arc  at  T  ;  connect  A  T  and  T  S.  On  B  as  centre  describe  the  arcs  K  W  and  E  V  ;  draw 
V  F  and  W  L  parallel  to  B  A  ;  make  V  F  and  W  L  equal  C  D  and  G  J  ;  through  S  L  F  A  trace  a  curve  on  the 
cutting  plane  that  will  lie  perpendicularly  over  the  plan  curve  A  D  J  P.  With  a  sharp-pointed  instrument 
scratch  the  lines  A  B  P  0  A  ;  cut  out  the  remainder  of  the  figure  and  touch  the  adjoining  edges  with  a  little 
glue  or  thick  mucilage  and  bring  them  together,  leaving  all  lines  on  the  outside  for  examination  and  study. 

Fig.  3.  Plan  of  Hand-rail  Less  than  a  Quarter-circle,  the  Tangents  to  the  Centre 
Curve  Line  A  P  Forming  the  Obtuse  Angle  P  B  A. — From  P  draw  P  M  and  P  5  at  right  angles  to 
B  P  ;  draw  A  5  at  right  angles  to  B  A  ;  on  5  as  centre  describe  the  curve  A  P.  The  position  of  the  tangent 
A  B  is  horizontal,  while  over  the  tangent  B  P  the  inclination  P  M  B  is  required.  Draw  T  0,  R  L,  U  G  and  X  C 
parallel  to  B  A  ;  parallel  to  P  M  draw  J  K,  F  I  E,  S  N  and  0  Q  ;  from  P  at  right  angles  to  B  A  draw  P  4  ;  on 
B  with  B  M  as  radius  describe  the  arc  M  4  :  then  4  A  will  be  the  distance  over  A  and  P  on  the  cutting  plane,  and 
if  a  line  be  drawn  from  4  to  B,  then  4  B  A  will  be  the  length  and  angle  of  tangents  on  the  cutting  plane.  To  find 
the  angle  for  squaring  the  wreath-piece  at  the  joint  over  P  :  draw  E  Z  parallel  to  B  P  ;  from  F  parallel  to 
B  M  draw  F  H  ;  draw  X  Y  at  right  angles  to  P  M  ;  make  X  Y  equal  P  H  ;  connect  Y  Z  :  then  the  bevel  at  Y 
will  give  a  plumb-line  on  the  butt-joint  over  P,  which  is  the  angle  sought.  To  find  the  angle  for  squaring  the 
wreath  at  the  joint  over  A  :  make  D  C  equal  J  K  ;  connect  C  A  ;  then  the  bevel  at  C  will  give  a  plumb-line  on 
the  butt-joint  and  the  angle  sought. 

Fig.  4.  Face-mould  Over  a  Plan  of  Less  than  a  Quarter-circle  with  One  Tangent 
Fixed  in  the  Horizontal  Plane,  the  Other  Inclined  as  Given  at  the  Plan  of  Hand-rail, 
Fig.  3. — Make  M  W  equal  A  4  of  Fig.  3  ;  with  B  M  of  Fig.  3  as  radius  set  one  foot  of  the  compasses  on  M 
and  ilescribe  an  arc  at  B  ;  on  W,  with  A  B  of  Fig.  3,  intersect  the  arc  at  B  ;  connect  W  B  and  B  M.  Make 
the  joints  W  and  M  at  right  angles  to  the  tangents.  Make  M  K  I  N  equal  M  K  I  N  of  Fig.  3  ;  through  K  I  N 
parallel  to  W  B  draw  C  L,  A  J  and  E  G  ;  make  K  G  equal  J  X,  and  K  E  equal  J  W  of  Fig.  3  ;  through  M  draw 
G  F  ;  make  M  F  equal  G  M  ;  make  I  J  and  I  A  ecjual  F  G  and  F  U  of  Fig.  3  ;  make  N  L  and  B  6  ecjual  S  L 
and  B  6  of  Fig.  3  ;  make  W  D  equal  W  C.  Through  G  J  L  6  D  on  the  convex  and  F  E  A  C  of  the  concave 
trace  the  curved  edges  of  the  face-mould. 

Fig.  5.  Parallel  Pattern  for  Round-rail  or  to  be  Used  Instead  of  the  Face-mould  as  a 
Means  of  Saving  Stuff,  and  for  Marking  the  Wreath-piece  on  the  Rough  Plank.— Make 
A  M  equal  A  4  of  Fig.  3  ;  make  the  tangents  M  B  and  B  A  equal  M  B  and  B  A  of  Fig.  3  ;  make  MFC  ecjual 
M  I  Q  of  Fig.  3  ;  make  F  V  and  0  T  equal  F  V  and  O  T  of  Fig.  3.  The  joints  are  at  right  angles  to  the  tan- 
gents.   On  M  V  T  and  A,  describe  circles  of  any  required  radius  for  width  of  pattern. 


*  Tangents  to  any  plan  curve  that  includes  less  than  a  quarter-circle,  or  a  curve  that  measures  less  than  ninety  degrees,  always 
form  obtuse  angles. 


Plate  No.  13. 


Plate  No.  14 


Fig.  4. 


PLATE  14. 


Fig.  I.  Represents  a  Solid  Block  or  Prism  Standing  Vertically  on  a  Base  A  B  C  D  the 
Sides  of  which  are  Parallel  and  Have  Two  Acute  and  Two  Obtuse  Angles. — The  upper  end 
of  this  prism  is  cut  on  the  inclination  C  E  B,  and  on  the  line  E  G  at  right  angles  to  the  sides  and  parallel  to 
the  base  C  D.  The  base  line  B  A  of  this  solid,  being  in  the  horizontal  plane  and  also  terminating  the  inclined 
plane,  is  a  level  line  common  to  both  planes.  On  the  base  draw  the  lines  A  Y  and  C  Y  at  right  angles  to  the 
tangents  ;  on  Y  as  centre  draw  the  plan  curve  A  L  H  C.  At  Plate  No.  13,  Fig.  i,  the  solid  is  precisely  like 
this,  but  the  obtuse  angled  tangents  to  the  curve  were  required  in  that  case,  because  the  plan  curve  was  less  than  a 
quarter-circle  ;  here,  however,  the  acute  angle  tnust  be  used  because  the  plan  curve  is  greater  than  a  quarter-circle* 
At  any  points  on  the  curve  as  L  and  H  parallel  to  A  B  draw  L  O  and  H  J  ;  parallel  to  C  E  draw  0  N  and  J  K; 
from  K  and  N  parallel  to  A  B  draw  N  M  and  K  F  ;  make  N  M  equal  0  L  and  K  F  equal  J  H  ;  through  the 
points  A  M  F  E  trace  a  curve  on  the  cutting  plane,  which  will  lie  perpendicularly  over  the  plan  curve  A  L  H  C. 
As  the  sides  of  the  solid  A  B  and  B  C  at  the  base  are  tangent  to  the  plan  curve,  so  A  B  and  B  E  are  tangent 
to  the  curve  traced  on  the  cutting  plane. 

Fig.  2.  Construction  of  a  Paper  Representation  of  the  Solid  with  its  Curved  Lines, 
Surfaces  and  Angles  as  Given  in  Perspective  and  Described  at  Fig.  i.— Let  ABC  D  be  the 
form  of  the  base,  the  opposite  sides  of  which  are  parallel  and  equal.  Draw  A  Y  and  C  Y  at  right  angles  to 
the  tangents  ;  on  Y  as  centre  describe  the  plan  curve  A  0  L  C.  Let  C  E  B  be  the  inclination  over  the  base 
C  B  ;  make  C  H  and  D  G  at  right  angles  to  C  D  and  equal  to  C  E  ;  connect  H  G  ;  make  D  I  at  right  angles 
to  A  D  and  equal  to  D  G  ;  connect  I  A.  On  B  with  B  E  for  radius  describe  the  arc  E  F  and  the  arc  K  ;  on  A 
as  centre  with  A  I  as  radius,  describe  an  arc  at  J  ;  on  A  as  centre  with  A  F  as  radius  intersect  the  arc  at  K  ; 
with  B  A  for  radius  on  K  intersect  the  arc  at  J  ;  connect  A  J,  J  K  and  K  B.  Parallel  to  A  B  from  any  point 
on  the  curve  0  and  L  draw  0  N  and  L  M  ;  parallel  to  C  E  draw  M  P  and  N  R  ;  make  BUT  equal  B  R  P  ; 
parallel  to  B  A  draw  U  Q  and  T  S  ;  make  U  Q  and  T  S  equal  N  0  and  M  L.  Through  K  S  Q  A  trace  a  curve 
on  the  cutting  plane  that  will  lie  perpendicularly  over  the  plan  curve  A  0  L  C.  With  a  sharp-pointed  instru- 
ment scratch  the  lines  A  B,  B  C,  C  D  and  D  A  ;  cut  out  the  remainder  of  the  figure  and  touch  the  adjoining 
edges  with  a  little  glue  or  thick  mucilage  and  bring  them  together,  leaving  all  lines  on  the  outside  for  com- 
parison and  study. 

Fig.  3.  Plan  of  Hand-rail  Greater  Than  a  Quarter-circle,  the  Tangents  to  the  Centre 
Curve  Line  A  C  Forming  the  Acute  Angle  A  B  C. — Draw  C  Y,  C  E  at  right  angles  to  C  B  ;  draw  A  Y 
at  right  angles  to  A  B  ;  on  Y  as  centre  describe  the  centre  curve  line  A  N  M  C.  The  tangent  A  B  is  to  remain 
level,  and  over  the  tangent  B  C  the  inclination  C  E  B  is  required.  Through  I  and  T  draw  I  R  and  T  V 
parallel  to  A  B  ;  at  any  point  on  the  curve  as  X  draw  K  G  parallel  to  A  B  ;  parallel  to  C  E  draw  Q  0,  K  L  and 
V  P.  From  C  at  right  angles  to  A  B  draw  C  F  indefinitely  ;  on  B  as  centre  with  B  E  as  radius  describe  the 
arc  E  F  :  then  A  F  will  be  the  distance  over  A  and  C  on  the  cutting  plane  ;  and  if  a  line  be  drawn  from  F  to  B, 
thenV  ^  k  will  be  the  length  and  angle  of  tangents  on  the  cutting  plane.  To  find  the  angle  for  squaring  the 
wreath-piece  at  the  joint  over  C  :  From  K  draw  K  H  parallel  to  B  E  ;  make  C  S  equal  C  H  ;  connect  S  J  : 
then  the  bevel  at  S  will  give  a  plumb  line  on  the  butt-joint  which  is  the  angle  sought.  To  find  the  angle  for 
squaring  the  wreath-piece  at  the  joint  over  A  :  make  Z  G  equal  K  L  ;  connect  G  A  :  then  the  bevel  at  G  will 
give  a  plumb  line  on  the  butt-joint  over  A  and  the  angle  sought.  In  finding  angles  for  squaring  wreath-pieces 
as  much  of  the  Joint  lines  as  are  convenient  may  be  taken,  as  follows  :  Prolong  the  Joint  line  C  Y  until  it  meets  the 
continuation  of  the  level  line  B  A  at  8,  rnake  C  6  equal  C  5  ;  connect  6,  8,  and  the  same  angle  will  be  given  at  C,  6,  8 
(Zi-  C  S  J  ;  and  again  at  Joint  A  ;  from  C  draw  the  line  C  7  parallel  to  B  k  ;  prolong  the  Joint  line  kV  to  2  ; 
make  2,  7  equal  C  E  :  connect -j  A  ;  then  the  angle  1,  1  k  equals  the  angle  Z  G  A. 

Fig.  4.  Face-mould  Over  a  Plan  of  Hand-rail  More  Than  a  Quarter-circle,  the  Plan 
Tangents  Forming  an  Acute  Angle,  One  of  the  Tangents  to  Remain  Level,  the  Other 
Inclined,  as  Given  at  the  Plan  Fig.  3. — Make  A  E  equal  A  F  of  Fig.  3,  with  E  B  of  Fig.  3  as  radius  ; 
set  one  foot  of  the  compasses  on  E  and  describe  an  arc  at  B  ;  on  A  with  A  B  of  Fig.  3  as  radius  intersect  the 
arc  at  B  ;  connect  E  B  and  A  B  ;  make  E  0  L  P  equal  E  0  L  P  of  Fig.  3.  Parallel  toA  B  draw  P  C,  L  D  and 
0  F.  0  G  ;  make  OG,  OF  equal  Ql  and  Q  R  of  Fig.  3;  through  E  draw  G  H;  make  EH  equal  EG;  make 
L  J  D  equal  K  3  X  of  Fig.  3;  make  P  K  C  equal  V  WT  of  Fig.  3;  make  B  M  equal  B  U  of  Fig.  3.  Tiie  joints 
E  and  A  are  at  right  angles  to  the  tangents.  Make  AN  equal  AC.  Through  the  points  H  FJ  K  M  N  on 
the  convex  and  C  D  G  of  the  concave  trace  the  curved  edges  of  the  face-mould. 

Fig.  5.  Parallel  Pattern  for  Round-rail,  or  to  be  Used  Instead  of  the  Face-mould  as 
a  Means  of  Saving  Stuff,  and  for  Marking  the  Wreath-piece  on  the  Rough  Plank.— A  E  B 
equals  A  F  B  of  Fig.  3  ;  E  L  P  equals  E  L  P  of  Fig.  3  ;  L  M  and  P  N  equals  K  M  and  V  N  of  Fig.  3.  The 
joints  are  at  right  angles  to  the  tangents.  On  E  M  N  A  as  centres  describe  circles  of  any  required  radius  for 
width  of  pattern. 


*  Tangents  to  any  plan  curve  that  includes  more  than  a  quarter-circle,  or  that  measures  more  than  ninety  degrees,  always  form 
acute  angles. 


PLATE  15. 


Fig.  I.  Represents  a  Solid  Block  or  Prism  Standing  Vertically  on  a  Base  A  B  C  D, 
the  Sides  of  Which  are  Equal  and  Parallel,  and  Have  Two  Acute  and  Two  Obtuse 
Angles. —  I'he  upper  end  of  tliis  prism  is  cut  on  the  angle  B  F  A  on  the  side  A  B  ;  and  on  tlie  side  B  C,  on 
the  same  angle  of  inclination  G  H  F  ;  therefore,  the  sides  of  this  solid  have  a  common  inclination,  and  a  line 
F  I  drawn  on  the  cutting  plane,  or  B  D  on  the  horizontal  plane,  is  a  level  line  common  to  both  planes.  On 
the  base  A  0  X  M  C  represents  a  plan  curve  less  than  a  quarter-circle,  to.  which  the  sides  A  B  and  B  C  of  the 
solid  represent  the  plan  tangents  ;  and  the  lines  A  F  and  F  H  represent  the  tangents  on  the  cutting  plane. 
To  find  the  trace  of  the  plan  curve  on  the  cutting  plane  at  any  points,  as  0  X  M,*  draw  0  J  and  N  M  parallel 
to  D  B  ;  parallel  to  B  F  draw  J  K  and  N  Q  ;  parallel  to  F  I  draw  Q  P  and  K  L  ;  make  Q  P  equal  N  M  and 
K  L  equal  J  O  ;  make  F  R  equal  B  X  ;  through  the  points  A  L  R  P  H  trace  a  curve  on  the  cutting  plane  which 
will  lie  perpendicularly  over  the  plan  curve  A  0  X  M  C. 

Fig.  2.  Construction  of  a  Paper  Representation  of  a  Solid  With  its  Surfaces  Curved 
Lines  and  Angles  as  Given  in  Perspective  and  Described  at  Fig.  i.— Let  A  B  C  D  be  the  form 
of  base  the  opposite  sides  of  which  are  parallel  and  equal.  Draw  C  X  and  A  X  at  right  angles  to  the  tangents  ; 
on  X  as  centre  describe  the  plan  curve  A  J  C.  At  right  angles  to  A  B  and  C  D  draw  B  F,  C  V  and  D  W  ;  at 
right  angles  to  A  D  and  B  C  draw  D  Y,  C  U  and  B  E.  Let  B  F  A  be  the  inclination  required  over  the  base — 
or  plan  tangents— A  B  and  B  C.  Make  B  E,  C  2,  2  U,  C  Z,  Z  V,  D  W  and  D  Y  all  equal  B  F  ;  connect  U  E, 
V  W  and  Y  A.  Through  C  and  A  draw  S  T  indefinitely  ;  on  B  as  centre  with  F  A  as  radius  describe  arcs  at 
S  and  T.  At  any  points  on  the  curve,  as  I  and  5,  draw  I  K  and  5  M  parallel  to  D  B  ;  parallel  to  B  E  draw 
M  N  ;  parallel  to  B  F  draw  K  L  ;  on  A  with  S  T  as  radius  describe  an  arc  at  H  ;  on  F  as  centre  with  D  B  the 
level  line  as  radius  describe  an  arc  at  I  ;  on  F  with  F  A  as  radius  intersect  the  arc  at  H  ;  on  A  as  centre  with 
A  F  as  radius  intersect  the  arc  at  I  ;  connect  A  I,  I  H,  H  F  and  F  I.  Make  F  R  equal  E  N  ;  parallel  to  F  I 
draw  R  Q  and  L  0  ;  make  L  0  equal  I  K,  F  P  equal  B  J,  and  R  Q  equal  M  5  ;  through  A  0  P  Q  H  trace 
a  curve  on  the  cutting  plane  that  will  lie  perpendicularly  over  the  plane  curve  A  J  C  at  the  base.  With  a 
sharp-pointed  instrument  scratch  the  lines  A  B  C  D  A  and  A  F  ;  cut  out  the  remainder  of  the  figure,  and  touch 
the  adjoining  edges  with  a  little  glue  or  thick  mucilage  and  bring  them  together,  leaving  all  lines  on  the  out- 
side so  that  their  connections  may  be  seen  and  studied. 

Fig.  3.  Plan  of  Hand-rail  Less  Than  a  Quarter-circle,  the  Tangents  A  B  and  B  C  to 
the  Centre  Curve  Line  A  C,  to  Have  a  Common  Angle  of  Inclination.— At  right  angles  to  B  C 
draw  C  H,  C  R  ;  at  right  angles  to  A  B  draw  A  R  ;  connect  R  B  :  then  R  B  is  the  direction  of  a  level  line 
common  to  both  planes  ;  let  C  H  B  be  the  inclination  assumed  or  required  over  the  tangent  C  B,  and  let 
B  V  A  be  the  same  angle  of  inclination  over  the  tangent  B  A  ;  B  V  being  at  right  angles  to  A  B  ;  through  E 
draw  E  G  parallel  to  R  B  ;  parallel  to  C  H  draw  J  K.  Through  A  C  draw  the  line  S  T  indefinitely  ;  on  B  as 
centre  with  B  H  as  radius  describe  the  arc  H  T  and  S  :  then  S  T  will  be  the  distance  over  C  and  A  on  the 
cutting  plane  ;  and  if  lines  be  drawn  from  Tand  S  to  B  :  then  the  lines  T  B  and  S  B  will  be  the  length  and 
angle  of  the  tangents  on  the  cutting  plane.  To  find  the  angle  for  squaring  the  wreath-piece  at  both  joints  :  con- 
tinue B  C  toY  indefinitely  ;  make  C  Y  equal  C  L  ;  connect  Y  R  :  then  C  Y  R  ivill  he  the  angle  required  and  the 
bevel  at  Y  will  give  a  plumb-line  on  the  butt-joints  of  the  7t<reath-piece  over  A  and  C. 

Fig.  4.  Face-mould  Over  a  Plan  of  Hand-rail  Less  Than  a  Quarter-circle,  the  Plan 
Tangents  Forming  an  Obtuse  Augle,  and  the  Inclination  of  Both  Tangents  Alike,  as  Given 
at  the  Plan  Fig  3. — Let  S  D  and  D  T  equal  S  D  and  D  T  of  Fig.  3  ;  draw  D  B  at  right  angles  to  S  T 
and  equal  to  D  B  of  Fig.  3  ;  connect  T  B  and  S  B.  Make  B  X,  B  0  equal  B  U,  B  Q  of  Fig.  3  ;  make  T  J, 
S  J  each  equal  H  K  of  Fig.  3  ;  through  J  and  J  draw  lines  at  right  angles  to  S  T  or  parallel  to  D  B';  make 
J  G  and  J  E  at  both  ends  equal  J  G  and  J  E  of  Fig.  3  ;  through  T  draw  E  Z  ;  make  T  Z  equal  T  E  ;  through 
S  draw  E  Z  ;  make  S  Z  equal  S  E.  The  joints  S  and  T  are  at  right  angles  to  the  tangents.  Through 
Z  G  0  G  Z  of  the  convex  and  E  X  E  of  the  concave  trace  the  curved  edges  of  the  face-mould. 


*  As  many  level  lines  may  be  drawn  on  the  plan  curve  for  measuring  trace  points  on  the  cutting  plane  for  face-moulds  as  seem 
desirable.  But  in  drawing  face-moulds,  certain  points  on  the  plan  must  always  be  taken  with  the  level  measuring  lines  ;  as,  for  in- 
stance, the  angle  of  tangents  B,  and  the  points  E,  both  of  Fig.  3. 


Plate  No. 15 


Plate  No.  16- 


FiG.  4. 


PLATE  16. 


Fig.  I.  Represents  a  Solid  Block  or  Prism  Standing  Vertically  on  a  Base  A  B  C  D 
the  Sides  of  which  are  Equal  and  Parallel  and  have  Two  Acute  and  Two  Obtuse 
Angles;  the  Upper  End  of  the  Solid  is  Shown  as  Cut  on  Two  Different  Angles:  the 
Side  A  B  on  the  Angle  B  M  A,  and  the  Side  B  C  on  the  Lesser  Angle  V  Q  M.*— On  the 
base  A  E  G  H  C  represents  a  plan  curve  less  than  a  quarter-circle,  to  which  the  sides  A  B  and 
B  C  of  the  solid  represent  the  plan  tangents,  and  the  lines  A  M  and  M  Q  represent  the 
tangents  on  the  cutting  plane.  Make  B  L  equal  D  S,  draw  LJ  parallel  to  A  B,  connect  J  S;  then 
J  S  will  be  a  level  line  on  the  cutting  plane.  Make  J  F  parallel  to  B  M,  connect  F  D;  then  FD 
will  be  the  direction  of  level  lines  on  the  horizontal  plane  common  to  both  planes.  At  any  points 
on  the  curve  at  the  base,  as  E  G  H.  draw  B  G  and  I  H  parallel  to  F  D;  parallel  to  B  M  draw 
I  P;  draw  P  0  and  M  N  parallel  to  J  S;  make  P  0  equal  I  H,  M  N  equal  B  G,  and  J  K  equal 
F  E;  through  A  K  N  0  Q  trace  a  curve  on  the  cutting  plane  which  will  lie  perpendicularly  over 
the  plan  curve  A  E  G  C. 

Fig.  2.  Construction  of  a  Paper  Representation  of  a  Solid  with  its  Angles,  Surfaces, 
and  Curved  Lines  as  Given  in  Perspective  and  Described  at  Fig.  i. — Let  A  B  C  D  be  the 

form  of  base  the  opposite  sides  of  which  are  parallel  and  equal.  Draw  A  T  and  C  T  at  right 
angles  to  the  tangents;  on  T  as  centre  describe  the  plan  curve  A  0  P  R  C.  At  right  angles  to 
A  B  and  CD  draw  B  H,  C  X,  and  D  V;  at  right  angles  to  B  C  and  A  D  draw  B  G,  C  F,  and  D  U; 
let  B  H  A  be  the  angle  of  inclination  required  over  the  base  or  plan  tangent  A  B;  make  B  G 
and  C  I  each  equal  B  H;  connect  G  I;  make  I  FG  the  angle  of  inclination  over  the  base  or  plan 
tangent  B  C,  I  F  to  be  less  in  height  than  B  H;  make  D  U,  C  W  and  D  V  each  equal  I  F;  connect  U  A; 
make  W  X  equal  B  H;  connect  XV.  Make  B  L  equal  D  U;  draw  L  M  parallel  to  B  A;  make  M  N  parallel 
to  B  H:  then  N  D  will  be  the  direction  of  level  lines  common  to  both  planes.  At  right  angles  to  D  N 
draw  CZ  and  A  Y  indefinitely;  on  B  as  centre  with  A  H  for  radius  describe  an  arc  at  Y,  and  again  on  B 
as  centre  with  G  F  for  radius  describe  an  arc  at  Z:  then  Y  Z  will  be  the  distance  over  A  and  C  on  the 
cutting  plane,  and  if  lines  are  drawn  from  Z  to  B  and  Y  to  B,  then  Z  B  and  Y  B  will  be  the  length 
and  angle  of  the  tangents  on  the  cutting  plane.  From  B  parallel  to  N  D  draw  B  P,  and  at  any 
point  on  the  curve,  as  R,  draw  R  Q  parallel  to  N  D.  On  H  as  centre  with  G  F  as  radius  describe 
an  arc  at  K;  on  A  as  centre  with  Y  Z  as  radius  describe  an  arc  at  K;  on  A  as  centre  with  A  U 
as  radius  describe  an  arc  at  J;  on  M  with  N  D  as  radius  intersect  the  arc  at  J;  connect  A  J,  J  K 
and  K  H;  make  H  E  equal  G  5;  parallel  to  M  J  draw  H  2  and  E  4;  make  M  S  equal  N  0,  H  2 
equal  B  P,  and  E4  equal  Q  R,  through  A  S  2  4  K  trace  a  curve  on  the  cutting  plane  that  will 
lie  perpendicularly  over  the  plan  curve  A  0  P  R  C  at  the  base.  With  a  sharp-pointed  instrument 
scratch  the  lines  A  B,  B  C,  C  D,  DA  and  H  A;  cut  out  the  remainder  of  the  figure  and  touch 
the  adjoining  edges  with  a  little  glue  or  thick  mucilage  and  bring  them  together,  leaving  all 
lines  on  the  outside  so  that  their  connections  may  be  seen  and  understood. 

Fig.  3.  Plan  of  Hand-rail  Less  than  a  Quarter-circle,  the  Tangents  to  have  Two 
Different  Angles  of  Inclination,  the  Angle  B  T  A  over  the  Tangent  A  B,  and  C  S  B  the 
Lesser  Angle  over  the  Tangent  C  B. — To  find  the  position  of  a  level  line  common  to  both 
planes:  draw  A  L  parallel  and  equal  to  B  C;  make  B  W  equal  C  S;  draw  W  X  parallel  to  B  A; 
make  X  U  parallel  to  B  T;  connect  U  L:  then  U  L  will  be  the  line  sought.  Parallel  to  U  L 
draw  P  Q,  B  M  and  G  H;  parallel  to  C  S  draw  0  R;  parallel  to  U  X  draw  F  4.  At  right  angles 
to  L  U  draw  C  D  and  A  E  indefinitely;  on  B  as  centre  with  B  S  as  radius  describe  the  arc  S  D, 
and  again  on  B  as  centre  with  T  A  as  radius  describe  an  arc  at  E:  then  D  E  will  be  the  distance 
over  A  and  C  on  the  cutting  plane;  and  if  lines  are  drawn  from  h  and  D  to  B,  D  B  and  E  B 
will  then  be  the  length  and  angle  of  tangents  on  the  cutting  plane. 

To  Find  the  Angle  for  Squaring  the  Wreath-piece  at  the  Joint  over  C:— Prolong 
the  tangent  B  C  to  N;  make  C  N  equal  C  R;  connect  N  M:  then  the  bevel  at  N  will  give  a 
plumb-line  on  the  butt-joint  over  C. 

To  Find  the  Angle  for  Squaring  the  Wreath-piece  at  the  Joint  over  A:— Prolong 
the  level  line  U  L  until  it  msets  the  continuation  of  the  joint-line  A  K  at  J ;  prolong  the  tangent 
B  A  to  i;  make  A  I  equal  U  V;  connect  I  J:  then  the  bevel  at  I  will  give  a  plumb-line  on  the 
butt-joint  over  A.    See  Plate  No.  74,  I'lc.  5. 

Fig.  4.  Face-mou'd  over  a  Plan  of  Hand-rad  Less  than  a  Quarter-circle,  with  Two 
Different  Angles  of  Inclination  over  the  Plan  Tangents. — Let  A  H  equal  E  D  of  Fig.  3 
Make  H  Z  equal  D  Z  of  Fig.  3.  On  Z  as  centre  with  Z  B  of  Fig.  3  as  radius  describe  an  arc 
at  B;  on  H  as  centre  with  S  B  of  Fig.  3  as  radius  intersect  the  arc  at  B;  connect  H  B  and  A  B, 
and  if  the  work  is  correct  A  B  will  equal  A  T  of  Fig.  3.  Connect  B  Z;  make  H  Q  equal  S  R  ot 
Fig.  3;  make  A  P  and  A  R  equal  A  4  and  A  X  of  Fig.  3;  through  Q  parallel  to  B  Z  draw  K  G, 
through  R  and  P  parallel  to  B  Z  draw  M  E  and  N  D;  make  Q  K  and  Q  G  equal  0  Q  and  0  P 
of  Fig.  3;  make  B  F  equal  B  2  of  Fig.  3;  make  R  M  and  R  E  equal  U  Y  and  U  3  of  Fig.  3; 
make  P  N  and  P  D  equal  F  H  and  F  G  of  Fig.  3;  through  A  draw  D  0;  make  A  0  equal  A  D; 
through  H  draw  G  I;  make  H  I  equal  H  G;  through  1  K  M  N  0  of  the  convex  and  G  F  E  D  of 
the  concave  trace  the  curved  edges  of  the  face-mould.  The  slide-line  on  a  face-mould  is  always 
drawn  at  right  angles  to  the  level  line. 


*  The  opposite  sides  of  all  the  solids  must  be  cut  on  parallel  angles  of  inclination. 


PLATE  17. 


Fig.  I.  Represents  a  Solid  Block  or  Prism  Standing  Vertically  on  a  Base  of  the 
Given  Form  A  B  C  D,  the  Parallel  Sides  of  which  are  Equal,  but  Two  of  the  Sides  are 
Larger  than  the  Other  Two.  The  Upper  End  of  the  Solid  is  Shown  as  Cut  on  Two 
Different  Angles  :  the  Side  A  B  on  the  Angle  of  Inclination  B  J  A,  and  the  Side  B  C  on  the 
Angle  of  Inclination  G  FJ. — On  ilie  base  AQ  PC  represents  a  plan  curve  elliptic  or  eccentric, 
to  which  the  sides  of  the  solid  AB  and  BC  represent  tlie  plan  tans^ents;  and  the  lines  A  J  and 
J  F  represent  the  tangents  on  the  cutting  plane.  Make  D  K  equal  B  J;  draw  K  L  parallel  to  A  D; 
connect  J  L:  then  J  L  will  be  a  level  line  on  the  cutting  plane.  Make  L  I  parallel  to  E  D; 
connect  I  B:  then  I  B  will  be  the  direction  of  level  lines  on  the  horizontal  plane  common  to  both 
planes.  At  any  point  on  the  curve  P  draw  PO  parallel  to  I  B;  draw  0  H  parallel  to  BJ;  make 
H  M  parallel  to  J  L  and  equal  to  0  P,  and  J  N  equal  B  Q;  through  A  N  M  F  trace  a  curve  on 
the  cutting  plane  which  will  lie  pei'pendicularly  over  the  plan  curve  A  Q  P  C. 

Fig.  2.  Construction  of  a  Paper  Representation  of  a  Solid  with  its  Angles,  Surfaces, 
and  Curved  Lines  as  Given  in  Perspective  and  Described  at  Fig.  i.— Let  A  B  C  D  be  the 
given  form  of  b.ise,  and  A  H  G  C  an  eccentric  or  elliptic  curve  lo  which  the  sides  A  B  and  B  C  of 
the  solid  are  tangent.  At  light  angles  to  A  B  draw  B  U,  C  N  and  D  M;  at  right  angles  to  B  C 
draw  B  E,  C  S  and  D  L;  let  B  U  A  be  the  angle  of  inclination  required  over  the  base  or  plan 
tangent  A  B;  make  B  E  and  C  R  equal  B  U ;  l<-t  R  S  E  be  the  angle  of  inclination  required  over 
the  base  or  plan  tangent  B  C;  make  D  L,  D  M  and  C  Q  each  equal  R  S;  connect  LA;  make  Q  N 
equal  B  U;  connect  N  M;  make  D  K  equal  B  U,  draw  J  I  parallel  to  D  L,  connect  I  B.  At  right 
angles  to  B  I  through  C  and  A  di'aw  A  P  and  C  0  indefinitely.  On  B  as  centre  with  E  S  as 
radius  describe  an  arc  at  0;  again  on  B  as  centre  with  U  A  as  radius  describe  an  arc  at  P: 
then  P  0  will  be  the  distance  over  A  and  C  on  the  cutting  plane.  On  U  as  centre  with  E  S  as 
radius  describe  an  arc  at  X.  On  A  as  centre  with  P  0  as  radius  intersect  the  arc  at  X;  on  A 
with  A  J  as  radius  describe  an  arc  at  V;  and  again  on  A  with  A  L  as  radius  describe  an  arc  at  W. 
On  U  as  centre  with  B  I  as  radius  intersect  the  arc  at  V;  connect  A  V  W,  W  X,  X  U  and  U  V; 
make  UY  equal  ET;  parallel  to  U  V  draw  Y2;  make  Y2  equal  FG,  and  UZ  equal  BH;  through 
A  Z  2  X  trace  a  curve  on  the  cutting  plane  that  will  lie  perpendicularly  over  the  plan  curve  A  H  G  C 
on  the  base.  With  a  sharp-pointed  instrument  scratch  the  lines  A  B  C  D  and  A  U;  cut  out  the 
remainder  of  the  figure  and  touch  the  adjoining  edges  with  a  little  glue  or  thick  mucilage  and 
bring  them  together,  leaving  all  lines  on  the  outside  so  that  their  connections  may  be  seen  and 
understood. 

Fig.  3.  Plan  of  Hand-rail,  an  Eccentric  or  Elliptic  Curve,  the  Tangents  of  Unequal 
Length  and  Two  Different  Angles  of  Inclination. — Let  C  P  B  and  B  Q  A  be  the  angles  of 
inclination  over  the  plan  tangents  C  B  and  B  A.  Make  A  L  parallel  and  equal  to  B  C;  make 
P  M  equal  B  Q;  parallel  to  C  B  draw  M  F;  parallel  to  C  P  draw  F  G;  connect  G  L:  then  G  L 
will  be  the  direction  of  a  level  line  common  to  both  planes.  Parallel  to  G  L  draw  J  K,  E  B  W 
and  U  S;  parallel  to  C  P  draw  I  N;  parallel  to  B  Q  draw  T  6.  At  right  angles  to  G  L  through 
C  and  A  draw  C  D  and  A  H  indefinitely.  On  B  as  centre  describe  the  arc  P  D;  again  on  B  as 
centre  with  QA  as  radius  describe  an  arc  at  H;  connect  H  D:  then  H  D  will  be  the  distance  on 
the  cutting  plane  over  A  and  C,  and  if  lines  are  drawn  from  D  and  H  to  B,  D  B  and  H  B  will 
then  be  the  length  and  angle  of  tangents  on  the  cutting  plane. 

To  Find  the  Angle  for  Squaring  the  Wreath-piece  at  the  Joint  over  C:— Prolong  the 
tangent  B  C  to  Z;  prolong  the  joint-line  C  J  until  it  meets  the  continuation  of  the  level  line  G  L 
at  Y;  make  C  Z  equal  M  0;  connect  Z  Y:  then  the  bevel  at  Z  will  give  a  plumbdine  on  the  butt- 
joint  over  C. 

To  Find  the  Angle  for  Squaring  the  Wreath-piece  at  the  Joint  over  A:  — Prolong  the 
joint-line  AU  until  it  meets  the  continuation  of  the  level  line  B  2  at  W;  prolong  the  tangent  BA 
to  X;   make  A  X  equal  B  R:   then  the  bevel  at  X  will  give  a  plumb-line  on  the  butt-joint  over  A. 

Fig.  4.  Face-mould  over  the  Plan  of  Hand-rail  Given  and  Described  at  Fig.  3.— Make 
A  B  equal  H  D  of  Fig.  3.  Make  BE  equal  D  5  of  Fi;;,  3.  On  E  as  centre  with  5B  of  FiG.  3  as 
radius  describe  an  arc  at  C;  on  B  with  B  P  of  Fig.  3  intersect  the  arc  at  C,  and  if  the  work  is 
correct  A  C  will  equal  A  Q  of  Fig.  3.  Connect  CEP,  B  C  and  A  C;  make  B  F  K  equal  P  N  F  of 
Fig.  3;  make  A  M  equal  A  6  of  Fig.  3;  parallel  to  E  C  draw  G  F  J,  QK  and  N  M  L;  make  FJ, 
FG  equal  I  K,  IJ  of  Fig.  3;  make  K  Q  --qual  G  3  of  Fig.  3;  make  C  D,  C  P  equal  E  B,  B  2  of  Fig.  3;  make 
M  N,  M  L  equal  T  S,  T  U  of  Fig.  3;  through  A  draw  L  0;  make  AO  equal  A  L;  through  B  draw 
G  h';  make  B  H  equal  BG;  through  H  J  D  N  0  of  the  convex  and  GQPL  of  the  concave  trace 
the  curved  edges  of  the  face-moidd.  The  joints  A  and  B  are  at  right  angles  to  the  tangents. 
The  slide-line  may  be  drawn  anywhere  on  the  face-mould,  but  must  always  be  made  at  right 
angles  to  the  level  lines.  It  is  a'  matter  of  convenience  to  draw  the  slide-line  from  the  centre  of 
either  joint  of  the  face-mould. 

Fig.  5.  To  Find  a  Common  Angle  of  Inclination  Over  Two  Different  Lengths  of 
Plan  Tangents  when  Required,  the  Total  Height  being  Given:— The  tangents  are  of  the 
same  length  and  angl<;  as  those  of  FiG.  3.  Let  C  D  equal  both  C  P  and  B  Q  of  FiG.  3.  Prolong 
C  B  indetinitelv;  on'  B  as  centre  with  B  A  as  radius  describe  the  arc  A  F;  connect  D  F;  parallel 
to  F  D  draw  B  E;  parallel  to  C  D  draw  B  H;  at  right  angles  to  A  B  draw  B  G;  make  B  G  equal 
B  H;  connect  G  A:  then  the  angle  of  inclination  B  G  A  is  the  same  as  the  angle  C  E  B.  In  this 
case  the  angle  for  squaring  the  wreath-piece  will  be  alike  for  both  joints. 


Plate  No.  17 


s 


Fig.  4- 


Plate  No. 18 . 


Fig.  4. 


PLATE  18. 


Fig.  I.  Represents  a  Solid  Block  or  Prism  Standing  Vertically  on  a  Base  of  the 
Given  Form  A  B  C  D,  the  Sides  of  which  are  Equal  and  Parallel  and  have  Two  Acute 
and  Two  Obtuse  Angles. — The  acute  angle  formed  by  the  position  of  the  sides  A  B,  B  C  of  the 
base  is  intended  in  this  case  to  represent  the  angle  of  plan  tangents,  embracing  in  the  plan 
a  curve  of  more  than  a  quarter-circle.  The  upper  end  of  the  solid  is  shown  as  cut  on  the  side 
A  B,  on  the  angle  of  inclination  B  F  A,  and  on  the  side  B  C,  on  the  same  angle  of  inclination 
M  G  F.  The  sides  of  this  solid  being  cut  on  a  common  angle  of  inclination,  the  heights  from 
the  base  D  E  and  B  F  are  alike,  and  therefore  a  line  drawn  on  the  cutting  plane  from  F  to 
E  will  be  a  level  line ;  and  at  the  base — or  horizontal  plane — a  line  drawn  from  D  to  B  will 
be  the  position  of  a  level  line  common  to  both  planes.  Over  the  base  A  B  and  B  C  the  lines 
A  F  and  F  G  represent  the  tangents  on  the  cutting  plane, — or  those  of  a  face-mould.  At  any 
points  on  the  curve  at  the  base  parallel  to  B  D  draw  Q  R  and  L  K  ;  through  R  and  K  par- 
allel to  B  F  draw  R  I  ;  parallel  to  F  E  draw  I  H  and  T  N  ;  make  T  N  equal  R  Q,  F  J  equal 
B  0,  and  I  H  equal  K  L ;  through  A  N  J  H  G  trace  a  curve  on  the  cutting  plane  which  will  lie 
perpendicularly  over  the  plan  curve  A  Q  0  L  C. 

Fig.  2.  Construction  of  a  Paper  Representation  of  a  Solid  with  its  Angles,  Surfaces, 
and  Curved  Lines  as  Given  in  Perspective  and  Described  at  Fig.  i. — Let  A  B  C  D  be  the 
form  of  base  the  opposite  sides  of  which  are  parallel  and  equal.  Draw  A  M  and  C  M  at  right 
angles  to  B  C  and  B  A  ;  on  M  as  centre  describe  the  plan  curve  A  C,  which  is  greater  than  a 
quarter-circle.  At  right  angles  to  A  B  draw  B  U,  C  F  and  D  I  ;  at  right  angles  to  B  C  draw  C  G, 
B  H  and  D  X  ;  let  D  X  A  and  B  U  A  be  the  common  angles  of  inclination  ;  make  B  H,  C  J,  J  G, 
C  4,  4  F  and  D  I  each  equal  B  U  ;  connect  G  H  and  F  I.  As  D  I  and  B  H  are  of  equal  heights,  a 
line  from  D  to  B  will  be  the  position  of  a  level  line  common  to  both  planes.  Through  C  A 
draw  the  line  P  E  indefinitely  ;  on  B  as  centre  with  U  A  as  radius  describe  an  arc  at  P  and 
at  E  ;  then  P  E  will  be  the  distance  over  A  and  C  on  the  cutting  plane.  At  any  points  on  the 
curve,  as  Q  and  R,  parallel  to  M  B  draw  Q  0  and  L  R  ;  parallel  to  C  G  draw  L  K  ;  parallel 
to  B  U  draw  0  S.  On  U  as  centre  with  P  E  as  radius  describe  an  arc  at  V  ;  on  A  as  centre 
with  A  X  as  radius  intersect  the  arc  at  V  ;  again,  on  V  as  centre  with  the  same  radius  describe 
an  arc  at  W  ;  on  U  with  H  G  as  radius  intersect  the  arc  at  W  ;  connect  A  V,  V  W,  W  U  and 
U  V  ;  make  U  2  equal  H  K  ;  parallel  to  U  V  draw  S  T  and  2  Z  ;  make  S  T  equal  Q  0,  U  Y 
equal  B  N,  and  2  Z  equal  L  R  ;  through  the  points  W  Z  Y  T  A  trace  a  curve  that  will  lie  per- 
pendicularly over  the  plan  curve  A  Q  N  R  C.  With  a  sharp  pointed  instrument  scratch  the 
lines  A  B  C  D  A  and  A  U  ;  cut  out  the  remainder  of  the  figure  and  touch  the  adjoining  edges 
with  a  little  glue  or  thick  mucilage  and  bring  them  together,  leaving  all  lines  on  the  outside, 
so  that  their  connections  may  be  seen  and  understood. 

Fig.  3.  Plan  of  Hand-rail  Greater  than  a  Quarter-circle  ;  the  Tangents  to  have  a 
Common  Angle  of  Inclination  C  F  B  over  the  Tangent  C  B,  and  B  K  A  over  the  Tangent 
B  A.  Through  A  and  C  draw  the  line  D  E  indefinitely  ;  on  B  as  centre  with  B  F  as  radius 
describe  the  arc  F  E  ;  and  again  on  B  with  the  same  radius  as  before  describe  an  arc  at  D  : 
then  D  E  will  be  the  distance  over  A  C  on  the  cutting  plane;  and  if  lines  are  drawn  from  D 
to  B  and  from  E  to  B,  then  D  B  and  E  B  will  be  the  length  and  angle  of  tangents  on  the 
cutting  plane.  J  B  will  be  the  direction  of  a  level  line  common  to  both  planes.  Parallel  to 
J  B  draw   L  0  and  C  P  ;  parallel  to  C  F  draw   N  G. 

To  Find  the  Angle  for  Squaring  the  Wreath-piece  at  Both  Joints :— Prolong  the  tan- 
gent B  C  to  H  ;  make  C  H  equal  C  G  ;  connect  H  J  :  then  the  bevel  at  H  will  give  a  plumb- 
line  on  the  butt  joints  over  A  and  C. 

Fig.  4.  Face-mould  over  a  Plan  of  Hand-rail  Greater  than  a  Quarter-circle,  the  Tan- 
gents having  a  Common  Angle  of  Inclination  as  given  at  the  Plan  Fig.  3. — Let  V  X  U 
equal  D  T  E  of  Fig.  3.  On  V  as  centre  with  B  F  of  Fig.  3  as  radius  describe  an  arc  at  W  ; 
and  on  U  as  centre  with  the  same  radius  as  before  intersect  the  arc  at  W  ;  connect  V  W,  U  W 
and  X  W  :  X  W  should  equal  T  B  of  Fig.  3.  Make  V  C  and  U  G  each  equal  F  G  of  Fig.  3  ;  par- 
allel to  W  X  draw  V  E,  C  B  A,  G  H  F  and  U  K  ;  make  V  E  and  U  K  each  equal  C  P  of  Fig.  3. 
Make  C  B  and  C  A  equal  N  0  and  N  L  of  Fig.  3  ;  and  again,  make  G  H  and  G  F  equal  N  0 
and  N  L  of  Fig.  3.  Through  U  draw  F  J  indefinitely  ;  through  V  draw  A  D  indefinitely  ;  make 
VD  equal  AV;  make  UJ  equal  U  F;  make  WZY  equal  B  Q  S  of  Fig.  3.  The  joints  V  and 
U  are  at  right  angles  to  the  tangents.  Through  DEBZHKJ  of  the  convex  and  A  Y  F  of  the 
concave  trace  the  curved  edges  of  the  face-mould. 

Fig.  5.  Parallel  Pattern  for  Round  Rail,  or  to  be  Used  Instead  of  the  Face-mould 
as  a  Means  of  Saving  the  Width  of  Stuff  for  Marking  the  Wreath-piece  on  the  Rough 
Plank.— Make  DTE  equal  D  T  E  of  Fig.  3  ;  make  E  B  and  D  B  each  equal  F  B  of  Fig.  3  ; 
connect  B  T,  B  D  and  B  E;  make  E  N  and  D  N  each  equal  F  G  of  Fig.  3;  make  N  M,  N  M 
each  equal  N  M  of  Fig.  3  ;  make  B  R  equal  B  R  of  Fig.  3.  Describe  circles  on  the  points 
E  M  R  M  D  as  centres  of  any  required  radius  for  width  of  pattern.  Bend  a  flexible  strip  of 
wood  or  other  material  by  which  to  mark  curve- lines  touching  the  circles  for  the  convex  and 
concave  edges  of  the  pattern.    The  joints  are  made  at  right  angles  to  the  tangents. 


PLATE  19. 


Fig.  I.  Represents  a  Solid  Block  or  Prism  Standing  Vertically  on  a  Base  of  the 
Given  Form  A  B  C  D,  the  Sides  of  which  are  Equal  and  Parallel  and  have  Two  Acute 
and  Two  Obtuse  Angles. — The  acute  angle  formed  l)y  the  position  of  the  sides  A  B,  B  C  of 
the  base  is  intended  in  this  case  to  represent  the  angle  of  plan  tangents  embracing  in  the  plan 
a  curve  of  more  than  a  quarter-circle.  The  upper  end  of  the  solid  is  shown  as  cut  on  the  side 
A  B  on  the  angle  of  inclination  B  F  A,  and  on  the  side  B  C  on  a  less  angle  of  inclination    M  G  T. 

To  Find  the  Position  of  a  Level  Line  on  the  Cutting  Plane  :— Make  B  N  equal  D  E; 
draw  N  H  parallel  to  A  B;  connect  H  E,  which  is  the  level  line  sought;  parallel  to  B  F  draw  H  J  : 
then  f/ie  line  D  J  on  the  horizontal  plane  will  be  the  position  of  a  level  line  coutmon  to  both  planes.  At 
any  points  on  the  curve  at  the  base  parallel  to  J  D  draw  0  R  and  L  K;  through  R  and  K 
parallel  to  B  F  draw  R  S  and  K  U;  parallel  to  H  E  draw  U  T  and  S  P;  make  U  T  equal  K  L, 
S  P  equal  R  0,  and  H  Q  equal  J  X;  through  the  points  A  P  Q  T  G  trace  a  curve  on  the  cutting 
plane  which  will  lie  perpendicularly  over  the  plan  curve  A  0  X  L  C. 

Fig.  2.  Construction  of  a  Paper  Representation  of  a  Solid  with  its  Angles,  Surfaces, 
and  Curved  Lines  as  Given  in  Perspective  and  Described  at  Fig.  i. — Let  A  B  C  D  be  the 
form  of  base  the  opposite  sides  of  which  are  parallel  and  equal.  C  X  and  A  X  are  at  right 
angles  to  A  B  and  B  C.  On  X  as  centre  describe  the  plan  curve  A  W  C,  which  is  greater  than  a 
quarter-circle.  At  right  angles  to  A  B  draw  B  E,  C  L  and  D  M ;  at  right  angles  to  B  C  draw  B  F, 
C  H  and  D  N;  let  B  E  A  be  the  angle  of  inclination  over  B  A;  make  B  F  and  CJ  each  equal  B  E; 
let  J  H  F  be  an  angle  of  inclination — over  C  B — less  than  B  E  A  over  B  A;  make  D  N,  D  M  and 
C  K  each  equal  J  H;  make  K  L  equal  B  E;  connect  L  M  and  N  A;  make  B  5  equal  D  N;  draw 
5  U  parallel  to  B  A,  and  U  V  parallel  to  E  B:  then  a  line  drawn  from  V  to  D  will  be  the  position 
of  a  level  line  common  to  both  planes.  At  right  angles  to  D  V  draw  A  Q  and  CP;  on  B  as 
centre  with  F  H  as  radius  describe  an  arc  at  P,  and  again  on  B  as  centre  with  E  A  as  radius 
describe  an  arc  at  Q:  then  Q  P  will  be  the  distance  over  A  C  on  the  cutting  plane.  At  any 
points  on  the  plan  curve  Y  and  J  draw  J  I  and  Y  0  parallel  to  D  V;  make  I  G  parallel  to  C  H ; 
make  V  U  and  0  Z  parallel  to  E  B;  on  U  as  centre  with  V  D  as  radius  describe  an  arc  at  R;  on 
A  with  A  N  as  radius  intersect  the  arc  at  R;  on  E  as  centre  with  F  H  as  mdius  describe  an  arc 
at  S;  on  A  as  centre  with  P  Q  as  radius  intersect  the  arc  at  S;  connect  A  R,  R  S,  S  E  and  U  R; 
make  E  T  equal  F  G;  parallel  to  U  R  draw  Z  2  and  T4;  make  Z  2  equal  0  Y,  U  3  equal  V  W,  and 
T4  equal  I  J;  through  the  points  S,  4,  3,  2  A  trace  a  curve  on  the  cutting  plane  that  will  lie 
perpendicularly  over  the  plan  curve  A  Y  W  J  C.  With  a  sharp-pointed  instrument  scratch  the  lines 
A  B  C  D  A  and  A  E;  cut  out  the  remainder  of  the  figure  and  touch  the  adjoining  edges  with  a 
little  glue  or  thick  mucilage  and  bring  them  together,  leaving  all  lines  on  the  outside,  so  that 
their  connections  may  be  seen  and  understood. 

Fig.  3.  Plan  of  Hand-rail  Greater  than  a  Quarter-circle,  the  Tangents  to  have  Two 
Different  Angles  of  Inclination:  BZA  over  the  Plan  Tangent  A  B,  and  a  Less  Angle 
of  Inclination  C  H  B  over  the  Plan  Tangent  C  B. — Make  A  4  parallel  and  equal  to  B  C; 
make  B  5  equal  C  H;  draw  5  L  parallel  to  A  B,  and  L  V  parallel  to  Z  B:  then  the  line  V4  will  be 
the  direction  of  a  level  line  common  to  both  planes.  Parallel  to  V  4  draw  S  U,  B  F  and  Q  X; 
draw  Y  N  parallel  to  Z  B,  and  TJ  parallel  to  C  H. 

To  Find  the  Angle  for  Squaring  the  Wreath-piece  over  the  Joint  C: — Prolong 
the  tangent  B  C  to  G;  make  C  G  equal  C  D;  connect  G  F;  then  the  bevel  at  G  will  give  a  plumb- 
line  on  the  butt-joint  over  C. 

To  Find  the  Angle  for  Squaring  the  Wreath-piece  over  the  Joint  A : — Make  E  R 
parallel  to  B  A  and  equal  to  V  M;  connect  R  A:  then  the  bevel  at  R  will  give  a  plumb-line  on 
the  butt-joint  over  A.  Through  C  and  A  at  right  angles  to  V  4  draw  C  I  and  A  K  indefinitely; 
on  B  with  B  H  as  radius  describe  the  arc  H  I;  and  again  on  B  as  centre  with  Z  A  as  radius 
describe  an  arc  at  K;  connect  K  I:  then  K  I  will  be  the  distance  over  A  and  C  on  the  cutting 
plane;  and  if  lines  are  drawn  from  K  to  B,  and  I  to  B,  then  I  B  and  K  B  will  be  the  length  and 
angle  of  tangents  on  the  cutting  plane  or,  which  is  the  same  thing,  of  the  face-mould. 

Fig.  4.  Face-mould  over  a  Plan  of  Hand-rail  Greater  than  a  Quarter-circle,  the 
Tangents  having  Two  Different  Angles  of  Inclination  as  Given  at  the  Plan  Fig.  3. — 
Make  H  C  A  equal  I  0  K  of  Fig.  3.  On  C  as  centre  with  0  B  of  Fig.  3  as  radius  describe  an  arc 
at  B;  on  H  as  centre  with  H  B  of  Fig.  3  as  radius  intersect  the  arc  at  B:  then  if  the  work  is 
correct  A  B  will  equal  A  Z  of  Fig.  3.  Connect  A  B,  B  H  and  C  B;  make  H  N  equal  H  J  of 
Fig.  3;  make  B  Q  K  equal  Z  L  N  of  Fig.  3;  parallel  to  C  B  draw  N  F,  N  0,  Q  L  and  K  J,  K  D; 
make  N  0,  N  F  equal  T  U,  T  S  of  Fig.  3;  make  Q  R  L  equal  V  W  P  of  Fig.  3;  make  K  J,  K  D 
equal  Y  X,  Y  Q  of  Fig.  3;  through  A  draw  D  E;  make  A  E  equal  A  D;  through  H  draw  F  G; 
make  H  G  equal  H  F.  The  joints  H  and  A  are  at  right  angles  to  the  tangents.  Through  G  0  RJ  E 
of  the  convex  and  F  L  D  of  the  concave  trace  the  curved  edges  of  the  face-mould.  7'he  slide- 
line  in  this  case  and  of  all  face-moulds  of  whatever  character  is  always  at  right  angles  to  the  level  line. 


Plate  No.  19 


G 


Plate  No.  20  ' 


PLATE   2  0. 

Skilful  workers  of  hand-rail  know  that  carefully  shaping  the  wood  next  the  joints  of  a 
wreath-piece  so  that  it  will  nicely  fall  ^n  with  its  adjoining  pieces,  of  whatever  character  they 
may  be,  is  of  the  first  importance  ;  next,  experience  has  taught  them  that  the  interval — the 
helical  surfaces — or  top  and  bottom  surfaces  between  joints,  compel  them  to  follow  certain 
curvatures  peculiar  to  each  kind  of  wreath-piece.  From  these  facts  of  experience  it  is  evident 
that  a  face-mould  is  not  only  a  means  of  shaping  the  sides  of  a  wreath  on  the  plane  of  the 
plank,  but  that  it  carries  with  it  certain  geometrical  curves  that  shape  the  top  and  bottom 
surfaces  of  the  wreath.  This  controlling  curve-line  of  helical  surfaces  of  wreaths  is  a  centre 
line,*  as  at  A  E  R,  Fig.  2,  and  is  found  in  the  development  of  the  central  cylindric  line  on  cut- 
ting planes  as  Z  J  U  of  Fig.  i,  Plate  No.  11,  and  C  0  Q  S  E  of  Fig.  i,  Plate  No.  10.  See  also 
the  similar  cases  of  the  two  last  solids  referred  to,  with  their  cutting  planes  brought  in  posi- 
tion over  the  plan  at  their  bases,  Fig.  6,  Plate  No.  ii.  A  round  hand-rail — controlled  from 
its  centre  as  it  properly  must  be — over  a  circular  or  curved  plan  affords  a  complete  demon- 
stration that  this  centre  cylindric  line  gives  shape  to  the  top  and  bottom  of  the  rail,  for 
while  its  curved  sides  hang  vertically  over  the  plan,  its  top  and  bottom  also  take  proper 
curves,  forming  its  own  casings  perfectly  suited  to  the  requirements  of  every  case.  To  meas- 
ure and  make  an  exact  drawing  of  this  centre  line  will  demonstrate  the  peculiar  form  of 
curvature  in  all  cases  of  wreaths,  and  also  show  exact  heights  at  every  point  over  an  eleva- 
tion of  treads  and  rises  embraced  within  any  curved  plan.  Thus  the  practical  use  of  develop- 
ing the  centre  line  will  be  to  get  the  length  of  balusters  in  any  position  as  required  on  wind- 
ing steps  in  cylinders  ;  also  the  ability  to  test  the  wreath-piece  over  the  elevation,  and 
determine  when  desirable  what  changes  to  make,  if  any,  in  the  inclination  of  tangents. 

Fig.  I.  The  Semicircle  ACM  is  a  Plan  of  the  Centre  Line  of  a  Hand-rail  with 
Tangents  to  Each  Quarter  A  B,  B  C  and  C  N,  N  M. — In  this  plan  the  two  quarters  that 
make  the  semicircle  are  of  the  same  character  as  the  first  two  solids  introduced.  The  first 
solid  at  Plate  No.  10  is  here  repeated  by  D  M  N  C  ;  the  other,  D  A  B  C,  at  Plate  No.  ii. 
At  Plate  No.  ii.  Fig.  6,  two  similar  solids  are  brought  together  showing  the  difference  in 
their  cutting  planes  as  placed  in  position  over  the  plan  of  a  semicircle  at  the  base.  The 
quarter-circle  A  C  over  its  tangents  A  B,  B  C  has  a  common  inclination  B  Y  A,  C  E  B  ;  also  the 
quarter-circle  M  C  over  one  tangent  C  N  has  the  same  inclination  N  R  C,  while  the  tangent 
M  N  remains  level.  On  the  quarter  A  C,  B  D  is  the  position  of  a  level  line  common  to  both 
planes,  as  before  shown  ;  and  on  the  quarter  C  M,  N  M  is  the  level  line  common  to  both 
planes.  Divide  the  quarter-circle  A  C,  and  the  quarter  C  M,  each  into  four  equal  parts  ;  through 
L  and  H  draw  LJ,  H  G  parallel  to  the  level  line  D  B;  parallel  to  C  E  draw  G  F;  parallel  to 
B  Y  draw  J  K  ;  through  0,  P,  Q  parallel  to  the  level  line   M  N  dtaw  0  S,  P  T  and  Q  U. 

Fig.  2.  Development  or  Unfolding  of  the  Centre  Line  of  a  Semicircular  Wreath  of 
Hand-rail  over  the  Plan  A  C  M  as  given  at  Fig.  i. — Draw  A  X  ;  make  A  L  I  equal  A  L  I  of 
Fig.  I.  Draw  I  Y  at  right  angles  to  AX;  make  I  Y  and  L  K  equal  J  K  and  B  Y  of  Fig.  t. 
Parallel  to  A  X  draw  Y  C  ;  make  Y  H  C  equal  I  H  C  of  Fig.  i  ;  draw  C  E  and  H  F  at  right 
angles  to  A  X  ;  make  H  F  and  C  E  equal  G  F  and  C  E  of  Fig.  i.  Parallel  to  A  X  draw  E  N  ; 
make  E  Q  P  0  N  equal  C  Q  P  0  M  of  Fig.  i.  Through  N  at  right  angles  to  A  X  draw  X  R  ; 
make  N  R,  0  S,  P  T  and  Q  U  at  right  angles  to  E  N  and  equal  to  N  R,  V  S,  W  T  and  X  U  of 
Fig.  I.    Through  the  points  AKYFEUTSR  trace  the  centre  line  sought. f 

Fig.  3.  The  Semicircle  Q  V  S  is  the  Plan  of  a  Centre  Line  of  Hand-rail  with  Tan- 
gents to  each  Quarter  Q  U,  U  V  and  V  W,  W  S. — Let  U  T  Q  be  the  inclination  required  over 
the  plan  tangent  Q  U,  and  V  Y  U  the  greater  inclination  required  over  the  plan  tangent  U  V  ; 
and  for  the  quarter  S  V  the  angle  WAV  must  be  the  same  as  V  Y  U  ;  the  plan  tangent  W  S 
remains  level.  To  find  the  level  line  for  the  quarter  Q  V,  let  Y  X  equal  U  T  ;  draw  X  Z  par- 
allel to  V  U,  and  Z  F  parallel  to  Y  V  ;  connect  F  R,  which  is  the  level  line  sought.  Divide  V  J 
in  two  parts  ;  parallel  to  F  R  draw  U  K  ;  divide  K  Q  into  three  parts  ;  divide  V  S  into  four 
parts  ;  parallel  to  the  level  line  S  W  draw  P  B,  0  C  and  N  D  ;  parallel  to  R  F  draw  H  G,  L  8 
and  M  6;  parallel  to  V  Y  draw  G  E;  parallel  to  U  T  draw  8,4  and  6,  5.  See  Plate  No.  12 
for  case  of  solid  like  R  Q  U  V  of  Fig.  3. 

Fig.  4.  Development  of  the  Centre  Line  of  a  Semicircular  Wreath  of  Hand-rail 
over  the  Plan  Q  V  S  as  given  at  Fig.  3.— On  the  line  Q  F  make  Q  M  L  K  equal  Q  M  L  K 
of  Fig.  3.  Erect  the  perpendiculars  K  T,  L  4  and  M  5  equal  to  U  T,  8,  4  and  6,  5  of  Fig.  3. 
Draw  T  V  parallel  to  Q  F  ;  make  T  J  H  V  equal  K  J  H  V  of  Fig.  3  ;  perpendicular  to  Q  F  draw 
V  Y,  H  E  and  J  Z  ;  make  V  Y,  H  E  and  J  Z  equal  V  Y,  G  E  and  F  Z  of  Fig.  3  ;  parallel  to  Q  F 
draw  YW  ;  make  Y  N  0  P  W  equal  V  N  0  P  S  of  Fig.  3  ;  through  W  at  right  angles  to  Q  F 
draw  F  A  ;  make  W  A,  P  B,  0  C  and  N  D  equal  W  A,  3  B,  2  C  and  I  D  of  Fig.  3.  Through  the 
points  Q54TZEYDCBA  trace  the  centre  line  sought. 

Fig.  5.  This  Portion  of  a  Circle  Less  than  a  Quarter  is  a  Plan  of  a  Centre  Line  of 
Hand-rail  with  its  Tangents  D  C  and  C  B. — Over  the  plan  tangent  DC  let  DEC  be  the 
angle  of  inclination  required,  the  tangent  C  B  to  remain  level.  Divide  the  curved  line  B  D 
into  any  number  of  equal  parts,  and  from  the  points  of  division  J  I  H  G  F  draw  lines  par- 
allel to  B  C  and  touching  D  C  ;  at  right  angles  to  D  C  draw  0  P,  N  Q,  M  R,  L  S  and   K  T. 

Fig  6.  Development  of  the  Centre  Line  of  a  Wreath-piece  over  a  Plan  of  a  Portion 
of  a  Circle  less  than  a  Quarter  as  given  at  Fig.  5. — On  the  line  D  B  make  D  F  G  H  1  J  B 
equal  the  spaces  between  the  corresponding  letters  at  Fig.  5.  Make  D  E,  F  T,  G  S,  H  R,  I  Q 
and  J  P  equal  D  E,  K  T,  L  S,  M  R,  N  Q  and  OP  of  Fig.  5.  Through  the  points  B  P  Q  R  S  T  E 
trace  the  centre  line  sought.     7'ke  solid  Fig.  5  is  given  in  detail  at  Plate  No.  13. 

*  The  tangents  are  invariably  placed  at  the  centre  of  the  width,  and  the  heights  and  angles  of  inclination  are 
also  always  at  the  centre  of  the  thickness  of  the  wreath-piece. 

f  A  practical  self-imposed  useful  lesson  would  be  to  make  the  two  solids  composing  Fig.  i  of  wood,  and  glu- 
ing them  together  in  the  manner  shown  at  FiG.  6,  Plate  No.  ii,  remove  the  wood  to  the  semicircle  at  the 
base  and  its  vertical  trace  on  the  cutting  planes  ;  then  wrap  Fig.  2,  the  development,  around  the  cylindric  surface 
of  Fig.    I,  and  test  the  curve  thus  obtained. 


PLATE  21. 


Fig.  I.  Plan  of  a  Centre  Line  of  Hand-rail  Greater  than  a  Quarter-circle ;  the 
Tangent  B  C  to  have  an  Angle  of  Inclination  Equal  to  C  D  B,  the  Other  Tangent  B  A  to 
Remain  Level. — Divide  the  circular  line  A  C  into  any  number  of  equal  jjarts,  and  from  each  of 
these  points  of  division  draw  lines  parallel  to  the  level  line  A  B  touching  the  tangent  B  C,  as 
0  V,  E  F,  G  H,  L  K  and  M  N;  parallel  to  C  D  draw  N  P,  K  J,  H  I,  F  S  and  V  W.  This  solid  is  given 
at  Plate  No.  14. 

Fig.  2.  Development  or  Unfolding  of  the  Centre  Line  of  a  Wreath-piece  over  a 
Plan  Greater  than  a  Quarter-circle  as  given  at  Fig.  i. — On  the  line  AC  make  AOEG  LMC 
equal  the  spaces  between  the  corresponding  letters  of  Fig.  1.  At  right  angles  to  A  C  draw  C  D 
equal  to  C  D  of  Fig.  i.  Make  M  P,  LJ,  G  l.  E  S  and  OW  parallel  to  C  D  and  equal  to  N  P,  K  J, 
H  I,  F  S  and  V  W  of  Fig.  i.    Through  A  W  S  I  J  P  D  trace  the  centre  line  soiigiit. 

Fig.  3.  Plan  of  a  Centre  Line  of  Hand-rail  Greater  than  a  Quarter-circle,  the 
Tangents  Q  T  and  T  X  to  have  the  Common  Angle  of  Inclination  X  Y  T  and  T  Z  Q  — 
Divitle  the  circular  line  Q  X  into  any  even  number  of  equal  parts,  aiui  fiom  each  of  these  points 
of  division  draw  lines  parallel  to  the  level  line  R  T  touching  the  tangents,  as  I  J,  G  H,  D  E  and 
A  B;  make  J  K  and  H  L  parallel  to  X  Y,  and  E  M  and  B  C  parallel  to  T  Z.  This  solid  is  given  at 
Plate  No.  18. 

Fig.  4.  Development  of  the  Centre  Line  of  a  Wreath-piece  over  a  Plan  Greater 
than  a  Quarter-circle  as  given  at  Fig.  3. — On  the  line  Q  P  let  Q  A  D  F  equal  the  spaces 
between  the  coriesponding  letters  of  Fig.  3.  At  right  angles  to  Q  F  draw  F  Z,  D  M  and  AC 
equal  to  T  Z,  E  M  and  B  C  of  Fig.  3.  Make  Z  X  parallel  to  Q  P;  make  Z  G  I  X  equal  F  G  I  X  of 
Fig.  3;  through  X  draw  Y  P  at  right  angles  to  Q  P;  make  I  K  and  G  L  parallel  to  X  Y;  make 
X  Y,  I  K  and  G  L  equal  X  Y.  J  K  and  H  L  of  Fig.  3;  through  QC  MZ  LK  Y  trace  the  curve  sought. 

Fig.  5.  Plan  of  a  Centre  Line  of  Hand-rail  Greater  than  a  Quarter-circle,  the 
Tangents  to  have  Different  Angles  of  Inclination;  the  Angle  D  C  F  over  the  Plan 
Tangent  D  F,  and  a  Less  Angle  of  Inclination  F  E  A  over  the  Plan  Tangent  A  F. — To  find 
a  level  line  common  to  l)oth  planes:  draw  A  B  ])ai-allel  and  equal  to  F  D;  make  C  L  equal  F  E; 
draw  L  J  parallel  to  D  F;  make  J  I  parallel  to  DC;  then  the  line  I  B  will  be  the  level  line  soi/glit. 
From  F  draw  F  P  parallel  to  I  B;  divide  P  A  and  0  D  each  in  two  equal  parts;  draw  Q  H  and 
N  M  parallel  to  I  B;  make  M  K  parallel  to  D  C,  ami  H  G  parallel  to  F  E.  This  solid  is  given  at 
Plate  No.  19. 

Fig.  6.  Development  of  the  Centre  Line  of  a  Wreath-piece  over  a  Plan  of  More 
than  a  Quarter-circle,  the  Tangents  having  Different  Angles  of  Inclination  as  given  at 
Fig.  5.  — On  the  line  A  R  make  A  Q  P  equal  A  Q  P  of  Fig.  5;  make  P  E  aiui  Q  G  perpendicidar 
to  A  R  and  equal  to  F  E  and  H  G  of  Fig.  5;  draw  E  D  parallel  to  A  R;  make  EO  N  D  equal 
POND  of  Fig.  5;  through  D  diaw  CR  at  right  angles  to  A  R;  draw  N  K  and  OJ  parallel  to 
DC;  make  D  C,  N  K  and  0  J  equal  D  C,  M  K'and  I  J  of  Fig.  5;  through  the  points  AGE  J  KC 
trace  the  centie-line  sought. 

Fig.  7.  Plan  of  a  Centre  Line  of  Hand-rail  Less  than  a  Quarter-circle,  the 
Tangents  S  U  and  U  V  to  have  the  Common  Angle  of  Inclination  V  X  U  and  U  W  S. — 
Make  ST  parallel  and  equal  to  U  V:  then  T  U  vvill  be  a  level  line  common  to  both  planes. 
Divide  the  curved  line  SV  into  anv  even  number  of  equal  parts  SADKGYV;  parallel  to  T  U 
draw  YZ,  G  H.  D  E  and  A  B;  parallel  to  VX  draw  Z  J  and  H  I;  make  E  F  and  B  C  parallel  to 
U  W.     7'///j  soliil  is  given  at  Plate  No.  15. 

Fig.  8.  Development  of  the  Centre  Line  of  a  Wreath-piece  over  a  Plan  of  Less 
than  a  Quarter-circle,  the  Tangents  having  a  Common  Angle  of  Inclination  as  given  at 
Fig.  7. — On  the  line  SL  make  S  A  D  K  equal  SA  D  K  of  Fig.  7;  at  right  angles  to  S  L  make 
K  W,  D  F  and  A  C  equal  U  W  E  F  and  B  C  of  Fig.  7;  draw  W  V  parallel  to  S  L;  make  WG  Y  V 
equal  K  G  YV  of  Fig.  7;  through  V  at  right  angles  to  S  L  draw  XL;  parallel  to  L  X  draw  Y  J 
and  G  I;  make  V  X,  YJ  and  G  T  equal  V  X,  Z  J  and  H  I  of  Fig.  7;  through  the  points  SCFWIJX 
trace  the  centre  line  sought. 

Fig.  9.  Plan  of  a  Centre  Line  of  Hand-rail  Less  than  a  Quarter-circle,  the  Tangents 
to  have  Different  Angles  of  Inclination  ;  P  R  M  over  the  Tangent,  M  P  and  N  Q  P  a  Less 
Angle  of  Inclination  over  the  Tangent  P  N. — Make  P  S  equal  N  Q,  and  ST  parallel  to  P  M; 
make  TV  parallel  to  R  P;  {\\)m  M  parallel  and  equal  to  P  N  draw  M  0:  then  OV  will  be  a  level 
line  common  to  botii  planes.  Divide  B  M  in  two  equal  parts  M  A  B;  draw  P  C  parallel  to  V  0; 
divide  C  N  into  three  equal  parts  C  D  E  N;  parallel  to  0  V  draw  E  F,  D  G  and  AU;  parallel  to  N  Q 
draw  F  I  and  G  H;  parallel  to  P  R  diaw  U  J.     Tin's  solid  is  given  at  Plate  No.  16. 

Fig.  10.  Development  of  the  Centre  Line  of  a  Wreath-piece  over  a  Plan  of  Less 
than  a  Quarter-circle,  the  Tangents  having  Different  Angles  of  Inclination  as  given  at 
Fig.  9. — On  the  line  M  F  make  M  A  B  C  equal  M  A  B  C  of  Fig.  9;  at  right  angles  to  M  F  make 
C  R.  B  T  and  A  J  equal  P  R,  V  T  and  U  J  of  Fig.  9;  draw  R  N  parallel  to  M  F;  make  R  D  E  N 
equal  C  D  E  N  of  Fig.  9;  through  N  (iraw  FQ  at  right  angles  to  M  F;  parallel  to  N  Q  draw  El 
and  D  H;  make  N  Q,  E  I  and  D  H  equal  N  Q,  F  I  and  G  H  of  Fig.  9;  through  the  points 
M  J  T  R  H  I  Q  trace  the  centre  line  sought. 

.V  further  important  use  for  a  knowledge  of  this  centre  line  is  to  unfcjld  side  moulds.  See  Plate 
No.  76,  Figs,  i,  2,  3,  4,  5,  and  6. 


Plate  No.  21 


M  A         B        C  F 

F 1 G. 10. 


Plate  No.  22  . 


PLATE  22. 

Position  of  Riser  in  Connection  with  Cylinders  at  the  Landing  and  Starting  of  Straight 
Flights  of  Stairs. — The  Face-mould,  and  the  Management  of  the  Wreath-pieces. 

Fig-.  I.  A  Sufficient  Elevation  of  Rises  and  Tread  to  Determine  the  Place  of  Riser 
at  the  Bottom  of  a  Flight  when  the  Over-wood  is  to  be  all  Removed  from  the  Top  of 
the  Wreath-piece,  as  in  this  Case. — Let  X  X  be  the  centre  of  the  short  balusters  and  the 
bottom  line  of  the  hand-rail  ;  make  X  B  the  thickness  of  rail,  and  X  C  =  3^-"  the  thickness  of 
plank  ;  draw  C  E  parallel  to  X  X  ;  make  X  A  half  the  thickness  of  plank  ;  draw  A  D  parallel 
to  X  X  ;  make  N  F  four  inches,  and  F  D  half  the  thickness  of  the  rail,  being  all  together,  from 
the  floor  to  S,  5I".  Where  the  centre  line  A  D  intersects  the  centre  level  line  S  D,  that  inter- 
section at  D  fixes  the  centre  of  wreath-piece  as  shown.  From  D  parallel  to  the  riser  draw 
D  G  indefinitely  ;  anywhere  below  the  floor-line  and  parallel  to  it  draw  K  G  ;  as  G  is  the 
centre  of  the  rail,  G  0  must  be  a  half-inch  to  the  face  of  the  cylinder  ;  and  as  the  diameter 
of  the  cylinder  is  6  ",  0  K  must  be  3"  ;  draw  K  L  parallel  to  the  riser  ;  continue  the  line  of 
the  first  riser  to  H  ;  on  K  with  K  0  for  radius  describe  the  cylinder  :  then  H  J  shows  the 
distance  to  be  i^"  between  the  bottom  riser  and  the  commencement  of  the  cylinder  ;  and  this 
is  so  placed  at  the  plan  Fig.  3,  as  may  be  seen  by  the  corresponding  letters. 

Fig.  2.  Elevation  of  Tread  and  Rise  at  the  Top  of  a  Flight  Sufficient  to  Determine, 
when  all  the  Over-wood  is  Removed  from  the  Bottom  of  a  Wreath-piece,  the  Relative 
Position  of  the  Riser  and  Cylinder. — At  the  bottom — Fig.  i — all  the  over-wood,  B  C,  is  taken 
off  the  top  of  the  straight  part  of  the  wreath-piece  ;  not,  however,  because  it  is  the  top,  but 
because  it  is  the  concave  face  ;  and  in  cases  of  this  kind  it  makes  the  best  shape  either  for 
the  top  or  the  bottom  of  the  flight.  At  the  top  of  the  flight  this  over-wood  is  taken  o&  at 
the  bottom  side  of  the  wreath-piece.  This  is  so  because  the  bottom  wreath-piece  is  simply 
turned  the  other  side  up  at  the  top  of  the  flight,  but  the  over-wood  is  still  taken  from  the 
concave  face.  The  position  of  the  cylinders  differ ;  at  the  bottom  of  the  flight  the  chord- 
line  of  the  cylinder  is  i^"  from  the  face  of  the  riser,  and  at  the  top  of  the  flight  it  is  2". 
The  letters  at  the  various  points  of  this  elevation  are  made  to  correspond  with  those  of  Fig.  i, 
so  that  having  examined  and  made  the  drawings  of  that  figure,  a  careful  inspection  of  this 
will  be  all  the  explanation  needed. 

Fig.  3.  Plan  of  the  Bottom  of  Flight  with  the  Riser  and  Cylinder  as  Determined 
at  Fig.  I. 

Fig.  4.  Plan  of  the  Top  Portion  of  Flight  with  the  Cylinder  and  Riser  Placed  as 
Determined  by  the  Trial  at  Fig.  2. — Of  the  plan  of  hand-rail  around  the  cylinder,  one 
quarter-circle  has  to  be  prepared  for  drawing  a  face-mould.  Q  X  and  X  G  are  the  plan  tan- 
gents to  the  centre  line  of  rail,  the  pitch-board  to  be  placed  as  shown  ;  continue  G  X  to  W  ; 
parallel  to  G  X  draw  A  V,  Z  T  and  K  Y  R. 

Fig  5.  Face-mould. — *  Draw  F  W  indefinitely  ;  make  W  V  T  R  equal  W  V  T  R  of  Fig.  4, 
At  right  angles  to  F  W  draw  W  B,  V  A,  T  0  Z  and  R  E  Y  ;  make  W  B  equal  X  G  of  Fig.  4. 
Through  B  draw  A  D  parallel  to  R  W  ;  make  B  D  equal  B  A  ;  make  B  C  equal  G  C  of  Fig.  4  ; 
make  T  0,  T  Z,  R  E,  R  Y  each  equal  S  0,  S  Z  and  Q  Y,  Q  Y  of  Fig.  4  ;  parallel  to  R  F  draw  Y  M 
and  E  L  ;  through  D  C  0  E  of  the  convex  and  A  Z  Y  of  the  concave  trace  the  curved  edges 
of  the  face-mould.  In  this  connection  is  shown  the  laying  out  of  joints  B  and  F  to  square 
the  wreath-piece. 

Fig.  6.  Perspective  Sketch  of  the  Wreath-piece,  showing  both  joints  prepared  for 
squaring,  and  the  application  of  the  face-mould  to  both  sides  of  the  stuff. 

Fig.  7.  Elevation  of  Tread  and  Rises  for  the  Top  and  Bottom  as  before  Given, 
the  Object  being  to  show  that  sometimes  by  a  Change  in  Removing  the  Over-wood 
the  Wreath-piece  may  be  Kept  in  its  Required  Position  as  to  Height  and  the  Chord- 
line  of  the  Cylinder  Brought  to  the  Face  of  the  Riser. — G  is  the  centre  of  the  rail,  G  0 
is  ^"  ;  OK  is  the  radius  of  the  cylinder.  The  height  from  the  floor  to  D  is  fixed  as  before, 
alike  at  the  bottom  and  at  the  top.  The  bottom  line  of  rail  must  pass  through  X  X,  the 
centres  of  the  short  balusters  ;  from  X  X  set  off  the  thickness  of  rail  2^";  and  from  D,  which 
must  be  at  the  centre  of  the  plank,  set  off  both  ways  half  the  thickness  of  plank  parallel  to 
X  X  :  this  will  show  how  the  over-wood  must  be  removed  from  the  straight  part  of  each  wreath- 
piece. f  T//e  bottom  of  the  rail  at  the  centre  of  the  wreath  is  kept  4"  above  the  floor  to  suit  the 
required  length  of  balusters  on  the  level.  At  X  X  the  short  balusters  are  2'.2"  from  the  top  of 
the  step  to  the  bottom  of  the  rail  ;  then  from  the  floor  to  the  bottom  of  the  level  rail  the 
height  will  be  4"  more,  equal  to  z' .6" ,  the  same  length  that  the  Icmgest  baluster  on  each  step 
has  to  be,  because  being  half  a  tread  back  from  the  short  baluster,  it  must  therefore  be  a 
half-rise  longer.  The  merchantable  lengths  of  ordinary  balusters  are  2'.4"  and  2  . 8",  thus  allow- 
ing one  inch  to  go  in  the  rail  and  one  inch  to  dovetail  in  the  step.  The  illustrations  given 
at  Figs,  i,  2  and  7  as  methods  of  working  wreath-pieces  and  disposing  of  the  over-wood  on  the 
straight  part  are  not  to  be  understood  as  applying  only  to  6''  cylinders;  for  after  fixing  the  required 
position  of  wreath-piece,  and  G  and  0,  then  0  K  may  be  any  radius  of  cylinder,  more  or  less. 
For  instance,  in  the  case  of  Fig.  i,  if  0  K,  instead  of  being  a  radius  of  3",  should  be  6",  then 
the  riser  would  set  i^"  into  the  12"  cylinder.  And  again,  if  instead  of  0  K  being  3",  it  should 
be  2"  radius,  then  there  would  be  2^"  straight  between  the  riser  and  the  chord-line  of  a 
4"  cylinder. 

NoTK. — It  must  be  understood  that  throughout  this  work,  with  the  exception  of  the  cases  given  at  Plates  34  and 
35,  all  joints  of  wreath-pieces  are  to  be  made  at  right  angles  to  their  tangents,  and  square  through  the  face  of  the 
plank. 

*  At  Plate  No.  10  are  given  perspective  and  geometrical  drawings,  and  the  formation  of  a  paper  representation  of 
a  solid,  a  plan  of  hand-rail,  and  from  it  a  drawing  of  a  face-mould,  which  together  give  a  complete  practical  knowl- 
edge of  the  application  and  drawing  of  face-moulds  of  this  kind. 

\  At  Fig.  7  about  all  that  can  be  done  further — if  desirable — within  the  limits  of  the  thickness  of  the  plank  will  be 
at  the  top  of  the  flight  to  take  all  the  over-wood  oft  the  top  of  the  wreath-piece,  and  at  the  bottom  of  the  tli^ht  take 
all  the  over-wood  oft  the  bottom  of  the  wreath-piece;  this  change  would  brmg  D  at  the  bottom  of  the  fligli!— ki cpiiig 
it  at  the  same  Viflnrht    about  i"  nearer  to  the  riser;  and   D  at  the  too.  about  J"  nearer  to  the  riser,  at  the  same  heiglit. 


PLATE  23. 

Platform  Stairs,  Half-turn,  or  Such  as  given  by  Plan  and  Elevation  at  Plate  No.  i.  Figs. 
3  and  4. — The  Position  of  Cylinder  with  the  Place  of  Connecting  Risers,  and  the 
Differences  Possible  by  Varying  the  Removal  of  Over-wood  from  the  Straight  Portion 
OF  Wreath-pieces;  also  How  to  Fix  the  Risers  Connecting  with  the  Cylinder,  so 
THAT  the  Whole  Wreath  may  have  One  Common  Inclination,  besides  Saving  Several 
Inches  of  Sikpimng-room. 

Fig.  I.  Elevation  of  Rises  and  Treads  Above  and  Below  the  Platform  to  Test  the 
Removal  of  Over-wood  from  the  Wreath-pieces. — The  bottom  line  of  rail  must  in  all  cases 
pass  through  the  centres  of  the  short  balusters  X  X.  As  the  rail  is  to  be  2\"  thick  by  4"  wide 
the  plank  out  of  which  the  wreath-pieces  are  to  be  worked  must  be  4"  thick.  Make  X  B  and 
B  C  each  2";  draw  B  D  and  C  E  parallel  to  X  X;  make  X  N  and  X  M  2V;  connect  N  M;  make 
R  U  half  a  rise,  3J";   make  U  D  half   the   thickness   of   rail,  make  D  F  and  D  S   each  2", 

half  the  thickness  of  plank;  make  X  J  2V,  the  thickness  of  rail;  draw  F  L,  J  P  and  S  Q  parallel 
to  X  X.  This  drawing  shows  that  keeping  the  rail  above  the  platform  at  the  height  R  U  and 
taking  all  the  over-wood  off  at  N  C  of  the  upper  wreath-piece,  the  lower  wreath-piece  must 
have  j"  of  over-wood  taken  off  at  J  L,  the  top,  and  1"  off  the  bottom,  X  Q.  From  D  parallel 
to  the  line  of  riser  draw  D  G;  at  right  .ingles  to  D  G  draw  G  K;  make  G  0,  i",  and  0  K,  3",  the 
radius  of  the  cylinder;  on  the  centre  K  describe  the  cylinder  Y  A;  parallel  to  D  G  through 
K  draw  Y  V,  and  continue  the  line  of  risers  to  T  Z:  then  A  T  will  lje  the  distance  between  the 
chord-line  of  the  cylinder  and  the  face  of  the  risers  at  the  platform.  Another  c/ianf^e  can  be 
made  by  taking  all  the  over-iuood  off  the  bottom  of  the  lower  wreath-piece,  and  this  would  bring  the 
riser  of  that  side  f"  further  from  the  chord-line  of  the  cylinder — all  together  2V. 

Fig.  2.  Elevation  the  Same  as  that  of  Fig.  I.— The  object  in  repeating  this  drawing 
is  to  call  attenti'in  to  still  another  change  in  removing  over-wood.  In  this  case  the  over-wood 
is  all  taken  off  the  bottom  of  the  upper  wreath-piece,  and  off  the  top  of  the  lower  wreath- 
piece,  bringing  the  chord-line  of  the  6"  cylinder  to  the  face  of  the  riser  as  shown.  An  inch 
variation  in  the  height  of  R  U  to  bring  the  over-wood  as  required  would  be  of  slight  importance. 
This  arrangement  of  wreath-pieces  and  over-wood  is  not  confined  to  any  size  of  cylinder;  for  instance, 
if  the  cylinder  is  to  be  12"  diameter,  then,  G  0  being  fixed  points,  0  K  would  be  6",  and  the 
risers  would  set  just  as  they  are,  but  would  be  in  the  cylinder  3".  At  FiG.  i  a  similar  change 
would  take  place  if  the  size  of  cylinder  is  altered;  that  is  to  say,  that,  G  0  being  fixed,  if 
the  radius  of  the  cylinder  is  made  less,  the  chord-line  would  be  drawn  nearer  0;  and  if  the 
radius  is  made  greater,  the  chord-line  of  the  cylinder  advances  further  towards  the  risers  and 
into  the  step,  as  the  increase  is  more  or  less.  The  face-mould  for  Figs,  i  and  2  is  to  be  found 
exactly  as  directed  at  Figs.  4  and  5  of  Plate  No.  22. 

Fig.  3.  Here  again  is  an  Elevation  of  the  Same  Tread  and  Rise  Connected  with  a 
Platform  and  the  Same  Size  Cylinder  as  given  at  Figs,  i  and  2. — This  elevation  is 
introduced  for  the  purpose  of  showing  how,  by  a  wholly  different  treatment,  the  wreath  in 
this  case,  or  indeed  of  any-sized  cylinder — within  reasonable  limits — connected  with  a  platform- 
stairs  as  given  by  plan  and  elevation  at  Plate  i.  Figs.  3  and  4,  may  be  carried  around  such 
cylinder  on  one  common  inclination,  saving  room  in  the  stepping,  and  making  a  superior  shaped 
wreath.  After  setting  up  the  elevation,  begin  by  drawing  the  bottom  lines  of  rail  through 
the  centres  of  short  balusters  at  X  X;  set  off  the  centre  line  and  the  thickness  of  rail  parallel 
to  XX  as  shown.  Draw  H  F  indefinitely;  make  H  N  and  N  F  each  equal  K  G,  3^',  the  radius 
of  a  6"  cylinder,  and  V'  more  to  the  centre  of  rail  and  baluster;  through  F  parallel  to  the 
riser-lines  draw  B  S;  through  N  draw  M  U;  at  M  and  J  parallel  to  H  F  draw  J  E  and  M  L: 
then  the  four  heights  C  E,  E  F,  F  L  and  L  B  will  be  equal;  anywhere  on  the  line  B  S  at  K 
as  centre  with  3"  =  K  0  as  radius  describe  the  cylinder,  and  with  V  more  radius  to  G  describe 
the  centre  line  of  rail  R  G  P;  draw  G  W,  P  Q  and  R  V  at  right  angles  to  PR;  make  the 
heights  Q  T,  G  W,  UV  and  RS  each  equal  J  N;  connect  S  U,  V  G,  W  Q  and  TP:  then  the 
four  heights  and  inclinations  are  set  in  proper  relation  to  their  base,  the  plan  tangents. 

Fig.  4.  Plan  of  Rail  the  Centre  Line  of  which  is  the  Quarter-circle  K  P  G  of  Fig  3. — The 
heights  and  inclinations  Q  T  P  and  G  W  Q  are  the  same  as  those  of  like  lettering  at  Fig.  3. 
Through  Q  draw  S  Z;  parallel  to  Q  Z  draw  F  E  and  P  B;  through  G  draw  P  K;  on  Q  as  centre 
draw  WK;  parallel  to  Q  T  draw  D  C. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  Both  Joints  :— Make 
G  Y  equal  G  X;  connect  Y  Z:  then  the  angle  as  taken  by  the  bevel  at  Y  will  give  a  plumb- 
line —  H  G  of  Fig.  5 — on  the  butt-joints,  by  means  of  which  the  wreath-piece  may  he  squared. 

Fig.  5.  Face-mould  taken  from  the  Plan  Fig.  4;  also  Showing  the  Squaring  of 
the  Wreath-piece  at  Both  Joints. — On  the  line  K  K  make  A  K,  A  K  each  equal  A  K  of  Fig.  4; 
make  A  T  at  right  angles  to  A  K  and  equal  to  A  Q  of  Fig.  4;  connect  K  T,  KT;  make  T  C,  T  C 
each  equal  T  C  'of  Fig.  4;  parallel  to  A  T  draw  KB,  KB,  C  E,  C  F,  C  E,  C  F;  make  K  B,  K  B 
each  equal  P  B  of  Fig.  4;  make  C  E,  C  F,  C  E,  OF  equal  D  E  and  D  F  of  Fig.  4;  make  T  S 
and  T  R  equal  Q  S  and  Q  R  of  Fig.  4;  through  K,  K  draw  the  lines  F  D,  F  D;  make  K  D, 
K  D  each  equal  F  K;  continue  T  K  to  L,  and  make  K  L  any  length  desirable,  or  equal  to  A  B 
or  C  D  of  Fig.  3;  parallel  to  K  L  draw  D  0  and  F  0;  make  the  joints  L  and  K  at  right 
angles  to  the  tangents.  Through  DBESEBD  of  the  convex  and  F  R  F  of  the  concave  trace 
the  edges  of  the  face-mould. 

Note.— At  Plate  No.  ii  are  given  perspective  and  geometrical  drawings  and  instruction  how  to  form  a  paper 
representation  of  a  solid,  also  a  quarter-circle  plan  of  hand-rait,  and  Irom  this  plan  a  drawing  of  a  face-mould;  and  at 
Fig.  7  of  that  Plate  an  explanation  of  the  line  K  G  A  P  of  Fic  4  of  this  Plate — wMch  together  g'lve  a  practical  knowledge 
of  the  application  and  drawing  of  face-moulds  of  this  kind.  Pirci  fions  in  di'tail  for  sliding  face-moulds  and  the  correct 
application  of  lievels  for  squaring  wreath-pieces  'will  lie  found  at  Pl.A  l'E  No.  56. 


Plate  No.  23 


^  Scale  IV2  In.=  1  Ft. 

s 


Plate  No.  24 


PLATE  24 


Figs.  I  and  2.  Starting  and  Landing  Elevations  Sufficient  to  Show  the  Position  in 
the  Cylinders  of  the  Starting  and  Landing  Risers. — The  cylinders  of  15"  diameter;  the  over- 
wood  to  be  removed  from  the  straight  part  of  wreath-piece  the  same  as  at  Figs,  i  and  2  of 
Plate  No.  22. 

Fig.  3.  Face-mould  from  Plan  of  Hand-rail  Fig.  2. — The  lettering  of  face-mould  and 
plan  are  alike;  and  as  face-mould,  plan,  and  elevations  all  have  been  before  carefully  explained 
at  Plate  No.  22  (the  whole  being  drawn  to  a  scale  of  i^"  to  the  foot),  it  will  not  be  necessary 
to  repeat  the  same  here. 

Where  the  diameter-line  of  a  large  cylinder  is  placed  at  the  face  of  a  landing-riser  it  will 
be  necessary  to  manage  the  case  as  explained  through  Figs.  4,  5  and  6.*  Just  here  attention 
may  as  well  be  called  to  the  important  fact  that  whenever  it  is  desirable  the  ivhole  radius  of  cylinders 
— such  as  are  introduced  in  this  Plate,  and  also  of  Figs,  i,  2  and  7  of  Plate  No.  22,  and  Figs,  i  and  2 
of  Plate  No.  23 — may  be  saved  or  used  for  step-room,  in  an  entirely  unobjectionable  and  workmanlike 
manner,  if  plan  and  wreathpiece  are  treated  as  directed  at  Plate  No.  33. 

Fig.  4.  Elevation  of  Tread  and  Rise  Sufficient  to  Take  Measurements  with  which 
to  Prepare  the  Plan  of  Hand-rail  for  the  Purpose  of  Drawing  a  Face-mould. — Draw  the 
bottom  line  of  rail  through  XX,  the  centres  of  short  balusters;  parallel  to  X  X  draw  ED,  the  centre 
of  the  thickness  of  rail;  make  R  U  =  4",  U  D  =  i:^";  from  T  draw  T  S  parallel  to  the  floor-line  ; 
from  D  at  right  angles  to  the  floor-line  draw  D  S. 

Fig.  5.  Plan  of  Hand-rail  with  the  Top  Riser  Placed  at  the  Diameter-line  of  a  15" 
Cylinder. — At  the  centre  of  the  width  of  rail  T,  and  at  right  angles  to  T  K,  draw  tlie  tangent 
TS  indefinitely;  from  D  of  Fig.  4  parallel  to  the  line  of  riser  draw  a  line  to  S  of  Fig.  5;  make 
SD  equal  SD  of  Fig.  4;  connect  DT;  from  S  draw  the  line  SMY  tangent  to  the  centre  line 
of  the  plan  of  rail;  from  K  at  right  angles  to  S  Y  draw  the  line  K  M:  then  M  will  be  the  joint 
of  the  wreath-piece,  and  the  remainder  of  the  rail  around  the  cylinder  from  M  to  I  will  be  level. 
Parallel  to  M  S  draw  N  J,  0  Q  and  T  W;  continue  M  S  to  A.  From  Tat  right  angles  to  S  M  draw 
the  line  T  Z  indefinitely;  on  S  with  D  T  as  radius  describe  an  arc  at  Z. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  M: — 
Make  M  Y  equal  S  D;   connect  Y  W:    then  the  bevel  at  Y  will  give  the  angle  required. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  over  the  Joint  T: — Prolong 
the  tangent  S  T  to  L;  make  T  L  equal  G  V,  and  connect  L  J :  then  the  bevel  at  L  will  give  the 
angle  required. f 

Fig.  6.  Face-mould  taken  from  the  Plan  Fig.  5,  also  Showing  the  Squaring  of  the 
Wreath-piece  at  Both  Joints.J — On  a  line  E  D  make  D  T  equal  D  T  of  Fig.  5.  T  E  may  be 
any  length  for  straight  wood.  On  T  as  centre  with  Z  M  of  Fig.  5  as  radius  describe  an  arc  at 
M;  on  D  as  centre  with  S  M  of  Fig.  5  as  radius  intersect  the  arc  at  M ;  connect  M  D  and  prolong 
M  D  to  S;  make  TCP  equal  T  C  F  of  Fig.  5;  parallel  to  M  D  tlirough  F,  C  and  T  draw  U  N, 
R  G  and  TB;  make  DS  equal  S  A  of  Fig.  5;  make  FU  and  F  N  equal  G  H  and  G  N  of  Fig.  5; 
make  CR  and  C  G  equal  BO  and  B  Q  of  Fig.  5;  make  TB  equal  T  P  of  Fig.  5;  through  T  draw 
RO;  make  TO  equal  T  R;  parallel  to  T  E  draw  R  H  and  0  A;  make  the  joints  E  and  M  at  right 
angles  to  the  tangents;  make  M  K  equal  M  N;  through  K  SU  R  of  the  convex  and  NG  B  0  of  the 
concave  trace  the  edges  of  the  face-mould.  The  bevel  at  Y,  showing  the  squaring  of  the  wreath- 
piece  at  joint  M,  is  taken  from  Y  of  Fig.  5;  and  the  bevel  at  L,  showing  the  squaring  of  the 
wreath-piece  at  joint  E,  is  taken  from  L  of  Fig.  5.    See  Plate  No.  74,  Figs,  i  and  2. 

*  The  method  of  proceeding  would  be  the  same  at  the  bottom,  or  starting,  of  a  staircase. 

f  The  angle  required  to  square  a  wreath-piece  at  each  joint  is,  in  every  case,  the  inclination  of  the  plane  of  the  plank 
along  the  joint  given  on  that  plane — which  is  the  joint  of  the  face-mould — and  a  plumb-line  on  the  butt-joint — which  is  a 
joint  made  square  from  the  face  of  the  plank  through  its  thickness. 

X  At  Plate  No.  13  are  given  perspective  and  geometrical  drawings,  and  instructions  how  to  form  a  paper  representation 
of  a  solid;  also  a  plan  of  hand-rail  less  than  a  quarter-circle,  and  from  this  plan  a  face-mould — which,  together,  give  a 
practical  knowledge  of  the  application  and  drawing  of  face-moulds  of  this  kind. 


PLATE  25. 

Fig.  I.  Plan  of  the  Top  Portion  of  a  Staircase  Winding  One  Quarter,  with  a  Small 
Cylinder  as  given  at  Plate  No.  5,  Fig.  5. — Let  H  be  the  centre  of  hand-rail  and  baluster ; 
on  A  as  centre  describe  the  centre  line  of  rail  H,  E,  N  ;  make  H  B,  B  M  and  M  N  tangents  to 
the  centre  line  of  hand-rail.  Space  the  balusters  around  the  centre  line  of  rail  and  on  the 
winders  and  steps  below  as  required.  Parallel  to  H  B  draw  A  F  and  M  L  indefinitely  ;  parallel 
to  A  B  from  0,  the  centre  of  baluster,  draw  0  S  ;  prolong  M  B  to  J  indefinitely  ;  parallel  to  B  J 
draw  S  G  indefinitely ;  parallel  to  M  L  from  the  centre  of  balusters  Q  and  C  draw  Q  K  and 
C  D  indefinitely.  Further  heights  and  inclinations  to  complete  this  drawing  will  be  obtained 
after  Fig.  2,  the  elevation,  is  set  up. 

Fig.  2.  Elevation  of  Tread  and  Rise  as  Figured,  and  as  Taken  from  Fig.  i,  the 
Plan. — This  elevation  is  set  up  for  the  purpose  of  finding  heights  and  inclinations  over  the 
plan  tangents  H  B,  B  E  and  E  M  of  Fig.  1  ;  also  through  the  development  of  the  centre  line 
H  G  D  K  L  to  find  the  exact  relation  of  the  wreath-piece  to  the  steps  and  rises,  and  by  this 
means  be  enabled  to  get  the  lengths  of  balusters  wherever  placed  on  the  centre  line  of  rail, 
around  the  plan  of  cylinder.  The  elevation  is  necessary,  too,  to  make  the  curve  and  fix  the 
length  and  joints  of  the  ramp  ;  also  to  get  the  odd  lengths  of  balusters  that  may  occur 
under  the  ramp  as  shown.  In  practice,  the  drawing  of  this  elevation,  and  of  such  elevations 
generally,  may  be  done  full  size,  if  desired,  very  conveniently  by  the  use  of  the  pitch-board, 
laying  its  hypothenuse  along  the  edge  of  a  drawing-board  ;  and  for  the  winders  transferring 
the  tread  and  rise  lines  from  the  pitch-board  by  the  use  of  a  long  parallel  straight-edge. 
The  treads  around  the  cylinder  must  be  measured  on  the  centre  line  of  rail  as  follows,  at 
the  plan  Fig.  i  :  From  H  to  1,  the  first  tread  in  the  cylinder,  take  its  measure  in  two  equal 
parts,  and  the  second  step  from  1  to  2  into  two  parts  :  then  2  N  of  the  remainder  of  the 
centre  line  is  on  the  line  of  the  floor.  After  completing  the  elevation  anywhere  along  the  line 
marked  chord-line,  set  off  H  B,  3I",  equal  to  H  B  of  Fig.  i.  Through  B  draw  B  J  indefi- 
nitely and  parallel  to  T  H  ;  at  any  point  along  the  line  B  J  set  off  J  E  equal  to  B  E  of  Fig.  i. 
Through  E  draw  E  F  parallel  to  B  J  ;  at  any  point  along  the  line  E  F  set  off  F  M  equal  to 
E  M  of  Fig.  i  ;  through  M  draw  the  line  M  L  indefinitely  ;  from  the  floor-line  to  L  set  up 
5I"  ;  then  L  becomes  a  fixed  point  from  which  the  line  L  R  may  be  drawn,  R  being  the 
centre  line  of  ramp  ;  R  may  be  raised  or  lowered  to  suit,  but  there  can  be  no  change  at  L. 
Wherever  the  line  L  R  cuts  the  lines  T  H,  B  J,  E  F  and  M  L,  as  at  H,  J  and  F,  draw  the  lines 
H  B,  J  E  and  F  M  parallel  to  the  lines  of  tread.  At  Fig.  i  make  B  J,  E  F  and  M  L  each 
equal  B  J  of  Fig.  2.  As  the  heights  are  alike,  connect  J  H,  F  B  and  L  E  of  the  last-mentioned 
figure.  Place  the  baluster  at  0  the  same  distance  from  the  chord-line  as  at  HO  of  Fig.  i, 
and  the  other  two  balusters  as  at  1  C  and  2  Q,  as  marked  alike  at  Figs,  i  and  2.  Through 
0,  C  and  Q  draw  0  G,  C  D  and  P  K  parallel  to  the  rises;  make  S  G,  C  D  and  P  K  equal  the 
heights  indicated  by  the  same  letters  at  Fig.  i.  Through  H  G  D  K  L  trace  the  centre  line  of 
wreath  ;  parallel  to  this  centre  line  trace  the  top  and  bottom  lines  of  the  wreath  as  shown 
by  the  short  dash-lines.  Place  the  centres  of  balusters  that  occur  under  the  ramp  as  at 
the  plan,  and  draw  the  dotted  lines  parallel  to  the  risers  ;  then,  to  find  the  length  of  any  of 
the  balusters  around  the  wreath  or  under  the  ramp ;  take  for  example  C  4  of  the  wreath,  which 
is  2f",  add  this  to  2'.2",  the  usual  height  of  balusters  at  X,  X,  then  the  baluster  at  C  will  be 
2'.4i"  at  its  centre  line    from  the  top  of  step  to  the  under  side  of  the  hand-rail  or  wreath. 

Fig.  3.  Plan  of  Hand-rail  from  the  Quarter-circle  H  E,  Fig.  i. — Make  the  heights  and 
angles  B  J  H  and  E  F  B  agree  with  the  corresponding  letters  at  Fig.  i  ;  draw  A  B  ;  from  T 
parallel  to  A  B  draw  T  U  ;  parallel  to  B  J  draw  0  G  ;  through  E  draw  H  Z  indefinitely  ;  on 
B  as  centre  with  B  F  as  radius  describe  the  arc  F  Z.  To  find  the  angle  with  which  to  square 
the  wreath-piece  at  both  joints,  prolong  B  E  to  R  indefinitely ;  make  E  R  equal  E  K ;  connect 
R  A  ;  then  the  bevel  at  R  will  give  the  angle  sought. 

Fig.  4.  Face-mould  from  Plan  Fig.  3. — On  a  line  Z  Z,  make  V  Z,  V  Z  each  equal  V  Z  of 
Fig.  3.  At  right  angles  to  Z  Z  draw  V  M  ;  make  V  M  equal  V  B  of  Fig.  3  ;  connect  M  Z,  M  Z 
and  prolong  M  Z  to  A ;  make  Z  A  equal  H  R  of  Fig.  2  ;  make  M  G,  M  G  each  equal  J  G  of 
Fig.  3  ;  through  G  and  G  parallel  to  V  M  draw  U  T,  U  T  ;  make  G  U,  G  U  each  equal  0  U  of 
Fig.  3  ;  make  G  T,  G  T  each  equal  0  T  of  Fig.  3  ;  make  V  S  equal  V  S  of  Fig.  3  ;  through 
Z  and  Z  draw  T  F,  T  F  ;  make  Z  F,  Z  F  equal  Z  T,  Z  T  ;  make  T  B  parallel  to  M  Z  ;  make  F  0 
and  T  D  paiallel  to  MA;  make  the  joints  A  and  Z  at  right  angles  to  the  tangents.  Through 
F  U  M  U  F  of  the  convex  and  T  S  T  of  the  concave  trace  the  edges  of  the  face-mould.  The 
squaring  of  the  wreath-piece  at  both  joints  is  shown  through  the  use  of  the  bevel  R,  R 
taken  from   R  of  Fig.  3. 

Fig.  5.  Plan  of  Hand-rail  from  the  Quarter-circle  E  N,  the  Tangents  E  M  and  N  M  of 
Fig.  I. — Make  the  angle  M  L  E  equal  that  given  at  M  L  E  of  Fig.  i  ;  parallel  to  M  L  draw 
U  0  and  V  Q. 

Fig.  6.  Face-mould  from  Plan  Fig.  5,  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints. — Draw  the  lines  M  F  and  M  X  at  right  angles  ;  make  M  N  equal  M  N 
of  Fig.  5  ;  make  the  straight  wood  N  X  from  2"  up,  at  pleasure.  At  right  angles  to  X  M 
through  X  and  N  draw  Z  U  and  D  E  ;  make  M  0  Q  F  equal  L  0  Q  E  of  Fig.  5.  Through  0,  Q 
and  F  parallel  to  X  M  draw  W  Y,  V  C  and  E  S  ;  make  F  Y,  F  W  equal  E  Y,  E  W  of  Fig.  5  ; 
make  Q  C  and  Q  V  equal  R  C  and  R  V  of  Fig.  5  ;  make  0  S  equal  Z  S  of  Fig.  5  ;  make  N  Z 
equal  N  U  ;  make  Z  D  parallel  to  N  X.  Through  Y  C  S  M  Z  of  the  convex  and  W  V  U  of  the 
concave  trace  the  edges  of  the  face-mould.  The  bevel  L  used  to  square  the  wreath-piece  at 
the  joint  X  is  from  L  of  Fig.  5.  The  case  of  face-tnould  Fig.  4  is  treated  in  detail  at  Plate 
No.  II,  and  face-tnould  Fig.  6  is  likewise  treated  at  Plate  No.  id.  The  development  of  the 
centre  line  of  the  wreath  H  G  D  K  L  of  Fig.  2  is  illustrated  and  explained  in  detail  at  Plate 
No.  20,  Figs,  i  and  2.  Sliding  face-moulds  to  plumb  the  sides  of  wreath-pieces,  also  direc- 
tions for  the  application  of  bevels  to  square  wreath-pieces,  is  given  at  Plate  No.  56. 


Plate  No. 25 


Plate  No.  26 


PLATE  26. 


Fig.  I.  Plan  of  the  Top  Portion  of  a  Staircase  Turning  One  Quarter  to  the  Landing 
with  Diminished  Steps  Around  the  Cylinder,  Curved  Rises,  and  Platform  as  given  at 
Plate  No.  5,  Fig.  6. — This  is  an  improved  plan  of  stairs  turning  one  quarter  without  the 
winders  as  by  the  old  method  given  at  Plate  No.  25,  but  the  treatment  of  the  hand-rail  is 
precisely  the  same  as  that  at  the  last-mentioned  Plate,  because  the  conditions  around  the 
cylinder  are  alike  ;  so  that  if  the  explanation  is  not  given  in  this  case  with  quite  as  much 
detail  as  before,  it  is  for  the  reasons  stated.  It  would  be  well  to  make  a  careful  study  of 
the  preceding  Plate  in  connection  with  this.  Let  F  C  A  be  the  centre  line  of  hand-rail  and 
F  D  C,  C  B  A  the  tangents.  Place  the  balusters  around  the  centre  line  of  hand-rail  as  shown. 
Prolong  F  D  to  G  indefinitely,  and  prolong  K  C  to  J  indefinitely  ;  also  D  B  to  N.  Connect 
K  B  ;  from  L  parallel  to  K  B  draw  L  0  ;  from  0  parallel  to  B  N  draw  0  M  indefinitely  ;  from 
E  parallel  to  F  G  draw  E  H  indefinitely.  The  necessary  heights  and  inclinations  to  complete 
this  drawing  will  be  established  by  drawing  the  elevation,  and  proceeding  as  directed  at 
Fig.  2.    See  Plafk  No.  77. 

Fig.  2.  Elevation  of  Treads  and  Rises  as  Figured  and  as  Taken  from  Fig.  i,  the 
Plan.— From  the  chord-line,  which  is  the  commencement  of  the  cylinder,  the  treads  are  to  be 
measured  on  the  centre  line  of  rail,  from  A  to  Z  in  two  parts,  and  from  Z  to  Q  in  two 
parts  ;  also  on  the  line  of  floor  Q  F  in  two  parts.  Measuring  the  treads  in  two  parts  is  done  to  get 
more  exactly  the  stretch-out  of  the  centre  line.  After  drawing  the  elevation  set  off  from  the  chord- 
line  the  length  of  tangent  A  B  of  Fig.  i  (which  in  this  case  is  3^")  three  times  as  shown, 
drawing  lines  at  each  of  these  distances  parallel  to  the  chord-line,  to  N  J  and  G  indefinitely  ; 
make  the  height  from  the  line  of  floor  to  G  equal  55" :  then  G  becomes  a  fixed  point  from 
which  the  line  G  R  may  be  drawn,  R  being  the  centre  line  of  ramp  it  may  be  raised  or 
lowered  to  suit,  but  no  change  can  be  made  at  G  without  changing  the  usual  length  of  bal- 
usters. Where  the  line  G  R  cuts  the  lines  I  J,  B  N  and  the  chord-line  as  at  J,  and  at  N  and  A,  draw 
the  lines  A  B,  N  I  and  J  D  parallel  to  the  lines  of  tread.  At  Fig.  i  make  B  N,  C  J  and  D  G 
each  equal  B  N  of  Fig.  2,  and  connect  G  C,  J  B  and  N  A.  Place  the  centre  of  balusters  L,  F 
and  E  as  on  the  plan  L,  C  and  E  Fig.  i,  measuring  from  each  riser  on  the  centre  line 
except  the  first  baluster,  which  is  measured  from  A,  the  chord,  to  L,  the  centre  of  baluster  ; 
parallel  to  the  rise  lines  through  E,  F  and  L  draw  P  H,  F  B  and  L  M  indefinitely;  make  P  H 
equal  P  H  of  Fig.  i  ;  make  C  B  equal  C  J  of  Y\g.  i  ;  make  0  M  equal  0  M  of  Fig.  i;  then 
through  G  H  B  M  A  trace  the  centre  line  of  wreath  ;*  parallel  to  this  centre  line  set  off  and  trace 
the  top  and  bottom  lines  of  the  wreath  as  shown  by  the  dotted  lines. 

To  Find  the  Length  of  any  Baluster  around  the  Wreath,  take  for  Example  : — F  K, 
which  is  3.y"  ,  add  this  to  2'.2"  the  height  of  balusters  at  X  X  :  then  the  baluster  at  F  will  be 
2' .<-^V  at  its  centre  line  from,  the  top  of  step  to  the  under  side  of  the  wreath. 

Fig.  3.  Plan  of  Hand-rail  from  the  Quarter-circle  C  F,  the  Tangents  C  D  and  F  D  of 
Fig.  I. —  Make  the  angle  D  G  C  equal  the  angle  D  G  C  of  Fig.  i.  From  R  and  S  parallel  to 
F  G  draw  R  Y  and  S  X. 

Fig.  4.  Face-mould  from  Plan  Fig.  3,  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints.f — Draw  the  lines  C  G  and  G  B  at  right  angles  ;  let  G  F  equal  D  F  of 
Fig.  3  ;  F  B  the  straight  wood  added  should  never  be  less  than  2";  through  F  and  B 
parallel  to  G  C  draw  R  A  and  D  C  indefinitely  ;  make  G  Y  X  C  equal  G  Y  X  C  of  Fig.  3;  through 
C  at  right  angles  to  G  C  draw  Z  I  ;  through  Y  and  X  parallel  to  F  G  draw  D  V  and  S  W 
indefinitely  ;  make  F  A  equal  F  R  ;  draw  A  C  parallel  to  F  B  ;  let  Y  V,  X  W  and  C  I  equal  U  V, 
T  W  and  C  I  of  Fig.  3  ;  let  X  S  and  C  Z  equal  T  S  and  C  Z  of  Fig.  3.  Through  A  G  VW  I  of 
the  convex  and  R  S  Z  of  the  concave  trace  the  edges  of  the  face-mould.  The  bevel  G  of  Fig.  3 
is  used  to  square  the  wreath-piece  at  the  joint  B  as  shown.  The  dotted  lines  show  the  least 
wood  required  to  form  the  wreath-piece. 

Fig.  5.  Plan  of  Hand-rail  from  the  Quarter-circle  A  C,  Tangents  C  B  and  A  B  of 
Fig.  I. — Let  the  angles  C  J  B  and  B  N  A  each  equal  B  N  A  of  Fig.  i.  Connect  P  B  ;  from  S 
draw  S  V  parallel  to  P  B  ;  from  U  draw  U  T  parallel  to  B  N  ;  through  A  C  draw  A  L  indefi- 
nitely ;  on  B  as  centre  with   B  J   as  radius  describe  the  arc  J  L 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  Both  Joints: — Prolong 
B  C  to  Q  indefinitely  ;  on  C  as  centre  with  C  W  as  radius  describe  the  arc  W  Q  ;  connect  Q  P: 
then  the  bevel  Q  will  give  the  angle  required. 

Fig.  6.  Face-mould  from  Plan  J  Fig.  5,  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints. — Let  C  J  and  C  A  each  equal  M  L  of  Fig.  5.  Draw  C  N  at  right 
angles  to  A  J  ;  make  C  N  equal  M  B  of  Fig.  5  ;  connect  N  J  and  N  A  ;  prolong  N  A  to  R  indefi- 
nitely ;  make  A  R  equal  A  R  of  Fig.  2  ;  make  the  joints  R  and  J  at  right  angles  to  the  tangents  ; 
let  CY  equal  M  R  of  Fig.  5;  make  TO.  TX  and  TO,  TX  both  equal  U  V  and  U  S  of  Fig.  5;  through  J 
draw  X  Z,  make  J  Z  equal  J  X  ;  through  A  draw  X  Z  ;  make  A  Z  equal  A  X  ;  parallel  to  A  R  draw 
Z  W  and  X  S  ;  parallel  to  N  J  draw  X  F  ;  through  Z  0  N  0  Z  of  the  convex  and  X  Y  X  of  the  con- 
cave trace  the  edges  of  the  face-mould.  The  bevel  Q  used  to  square  the  wreath-piece  at  the 
joints  J  and  R  is  taken  from  Q  of  Fig.  5. 


*  The  development  of  the  centre  line  of  a  wreath,  as  in  this  case,  is  illustrated  and  explained  in  detail  at  Plate  20, 
Figs,  i  and  2. 

Sliding  face-moulds  to  plumb  the  sides  of  wreath-pieces,  also  directions  for  the  application  of  bevels  to  square 
wreath-pieces,  are  given  at  Plate  No.  56. 

f  The  face-mould  Fig.  4  is  explained  in  detail  at  Plate  No.  10. 
\  Face-mould  Fig.  6  is  explained  in  detail  at  Plate  No.  ii. 


PLATE  27 


Fig.  I.  Plan  of  the  Bottom  or  Starting  Portion  of  a  Staircase  Winding  One  Quarter, 
Similar  to  Plan  Fig.  I  of  Plate  No.  4.— Let  A  C  F  be  the  centre  line  of  rail  around  the 
cylinder,  and  A  B,  B  D  and  D  F  tangents.  Prolong  F  A  to  Q  indefinitely;  prolong  A  B  to  N  and 
H  C  to  T  indefinitely.  Space  the  balusters  on  the  centre  line  as  shown.  Further  heights  and 
inclinations  necessary  to  complete  this  drawing  will  be  obtained  as  directed  from  the  elevation 
Fig.  2,  when  that  drawing  is  completed. 

Fig.  2.  Elevation  of  Tread  and  Rises  as  Figured  and  as  Taken  from  Fig.  i,  the 
Plan. — Besides  finding  heights  and  inclinations  over  tangents,  this  elevation  is  also  set  up  to 
develop  the  centre  line  of  wreath  D  U  L  P  Q  in  its  exact  relation  to  the  step  and  rise,  thus 
giving  the  lengths  of  balusters  under  the  wreath.  The  treads  in  the  cylinder  must  be  measured 
around  the  centre  line  of  rail,  and  each  tread  taken  in  two  parts  in  order  to  get  more  exactly 
the  stretch-out  of  the  circular  line.  After  completing  the  elevation,  anywhere  along  the  line 
marked  chord-line, — which  is  the  commencement  of  the  cylinder, — set  off  H  A  or  A  B  of  Fig.  i 
(either  of  which  is  5^')  three  times,  drawing  lines  parallel  to  the  chord-line  indefinitely  as  shown. 
Let  X  X  at  the  centres  of  the  short  balusters  be  the  bottom  line  of  rail;  draw  R  N,  the  centre 
line,  indefinitely  and  parallel  to  XX;  at  the  intersection,  N,  draw  N  A  at  right  angles  to  the 
chord-line;  make  Q  R.  3",  for  straight  wood  to  be  added  to  that  end  of  wreath-piece.  Make 
E  D,  si"'^  connect  D  N;  at  T  and  D  draw  the  lines  D  C  and  T  B  parallel  to  the  treads.  At 
Fig.  I  make  CT  and  B  N  each  equal  CT  and  B  N  of  Fig.  2;  let  AQ  equal  AQ  of  Fig.  2; 
connect  Q  B,  N  C  and  T  D;  make  BO  equal  A  Q;  parallel  to  B  C  draw  OM;  parallel  to  B  0 
draw  M  G;  connect  G  H,  the  level  line  common  to  both  planes;  parallel  to  G  H  draw  J  K  and 
S  R;  parallel  to  A  Q  draw  R  P;  parallel  to  T  C  draw  K  L  and  Z  U.  At  Fig.  2  the  centre  of 
balusters  EZJ  and  S  are  placed  on  each  tread  as  at  the  plan,  and  the  lines  S  P,  J  L  and  VZU 
are  drawn  indefinitely  and  parallel  to  the  rise-lines;  make  V  U  and  K  L  equal  V  U  and  K  L  of 
Fig.  i;  make  G  P  equal  R  P  of  Fig.  i;  through  tlie  points  DULPQ  trace  the  centre  line  of 
wreath;  the  doited  lines  are  the  top  and  bottom  of  wreath  set  off  from  the  centre  line.  D  H 
is  straight  wood  that  will  be  added  to  that  end  of  the  lower  wreath-piece. 

To  Find  the  Lengths  of  Balusters  Around  the  Wreath  : — Take  for  example  the  baluster 
at  S;  S  F  is  5^",  which,  added  to  the  usual  height  of  balusters  at  X  X,  2'. 2",  makes  the  height 
of  this  baluster  on  its  centre  line  from  the  top  of  step  to  under-side  of  wreath  2'.jl". 

Fig.  3.  Plan  of  Hand-rail  from  the  Quarter-circle  F  C  of  Fig.  i  with  the  Tangents 
F  D  and  D  C— Make  CTD  equal  CTD  of  Fig.  i;    parallel  to  F  D  draw  P  B  and  Q  A. 

Fig.  4.  Face-mould  from  Plan  Fig.  3 ;  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints. — Ma.ie  the  lines  ND  and  Dl  at  right  angles;  let  DBAN  equal  DBAT 
of  Fig.  3;  let  D  F  equal  D  F  of  Fig.  3,  and  Fl  equal  DH  of  Fig.  2;  parallel  to  N  D  through 
F  and  I  draw  K  M  and  J  L;  through  B,  A  and  N  draw  R  L,  0  H  and  E  G  parallel  to  D  F;  make 
D  J,  B  R,  A  0,  N  E  and  N  G  each  equal  D  J,  K  R,  X  0  and  C  E  of  Fig.  3;  make  A  H  equal  XQ 
of  Fig.  3;  make  FK  equal  FM;  make  KJ  parallel  to  Fl;  through  M  H  G  of  the  concave  and 
KJ  RO  E  of  the  convex  trace  the  edges  of  the  face-mould.  TAe  bevel  at  T  used  to  square  the 
wreath-piece  at  Joint  1  is  taken  fromT  of  ¥\G.  3.  The  dotted  lines  show  the  width  of  wood  required 
to  work  out  the  wreath-piece.     This  face-mould  is  explained  in  detail  at  Plate  No.  10. 

Fig.  5.  Plan  of  Hand-rail  from  the  Quarter-circle  A  C  of  Fig.  i  with  the  Tangents 
AB  and  C  B. — The  angles  of  inclination  BNC  and  AQB  are  taken  from  Fig.  i.  Make  BO 
equal  A  Q;  make  0  6  parallel  to  B  C,  and  6S  parallel  to  0  B;  connect  S  H,  the  level  line  common 
to  both  planes.  Parallel  to  S  H  draw  Y  R,  5  L,  B  G,  T  U  and  AV;  parallel  to  B  0  draw  4  M 
and  R  P;  parallel  to  A  Q  draw  U  W;  at  right  angles  to  H  S  draw  AE  indefinitely;  draw  C  K  at 
right  angles  to  H  S;  on  B  as  centre  with  N  C  as  radius  describe  an  arc  at  K;  again  on  B  as 
centre  with  B  Q  as  radius  describe  the  arc  QE;  connect  E  K. 

To  Find  the  Angles  with  which  to  Square  the  Wreath-piece : — Prolong  B  A  to  F 
indefinitely  and  AH  to  J  indefinitely;  make  A  F  equal  A  D;  connect  F  G:  then  the  bevel  at  F 
will  square  the  wreath-piece  over  the  joint  A.  Make  H  J  equal  SX;  connect  J  C:  then  the  bevel 
at  J   will  square  the  wreath-piece  over  the  joint  C. 

Fig.  6.  Face-mould  from  Plan  Fig.  5;  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints. — Draw  the  line  K  Q;  let  ZQ  and  ZK  equal  ZE  and  ZK  of  Fig.  5.  On  Z 
as  centre  with  Z  B  of  Fig.  5  as  radius  describe  an  arc  at  B;  make  Q  B  equal  Q  B  of  FiG.  5,  and 
K  B  equal  C  N  of  Fig.  5;  connect  K  B,  B  Q  and  BZ;  make  BW  equal  B  W  of  Fig.  5;  make 
B  M6  P  equal  N  M6  P  of  Fig.  5;  parallel  to  Z  B  draw  Q  V,  M  L  5,  6,  3,  1  and  PY:  make  Q  V  and 
WT  equal  AV  and  U  T  of  Fig.  5;  make  Z  2,  M  L  and  M5  equal  Z  2,  4,  5  and  4  L  of  Fig.  5;  make 
6,  3,  6,  1,  PY  equal  S  I,  S  3  and  R  Y  of  Fig.  5;  through  Q  draw  the  line  T  C;  make  QC  equal  QT; 
through  K  draw  the  line  YA;  make  KA  equal  KY;  parallel  to  B  E  dra^w  TD  indefinitely;  make 
Q  E  equal  Q  R  of  Fig.  2.  The  joints  E  and  K  are  made  at  right  angles  to  the  tangents.  Through 
A3  L2VC  of  the  convex  and  YI5T  of  the  concave  trace  the  edges  of  the  face-mould.  The  slide- 
line  is  drawn  at  right  angles  to  B  Z.  The  dotted  lines  show  the  least  width  of  wood  required  to 
work  out  the  wreath-piece.  This  face-mould  is  explained  in  detail  at  Plate  No.  12.  The 
development  of  the  centre  line  of  this  case  of  wreath  is  given  in  detail  at  Plate  No.  20,  Figs. 
3  and  4.  Sliding  face-moulds  to  plumb  the  sides  of  wreath-pieces,  also  direction  for  the  application 
of  bevels  to  square  wreath-pieces,  is  given  at  Plate  No.  56. 


Plate  No.  2  / 


Plate  No. 28 


PLATE  28. 


Fig-.  I.  Plan  of  the  Top  Portion  of  a  Winding  Staircase  Making  a  Half-turn  with 
a  id"  Cylinder  as  given  at  Plate  No.  6,  Fig.  2. — Let  A  C  B  be  the  centre  line  of  rail  around 
the  plan  of  cylinder,  and  A  E,  E  D  and  D  B  tangents.  Prolong  E  D  to  T  indefinitely  ;  prolong 
F  C  to  J,  A  E  to  L,  and  FA  to  R,  all  indefinitely.  Space  the  balusters  on  the  centre  line  as 
shown.  Further  heights  and  inclination  of  tangents  to  be  shown  in  connection  with  this  plan, 
including  measurements  that  are  necessary  to  develop  the  centre  line  of  wreath  and  at  the 
same  time  fix  the  lengths  of  balusters,  will  be  obtained  afier  drawing  the  elevation. 

Fig.  2.  Elevation  of  Treads  and  Rises  as  Figured  and  as  Taken  from  Fig.  i,  the 
Plan. — The  treads  in  the  cylinder  must  be  measured  around  the  centre  line  of  rail,  and  each 
tread  taken  in  two  parts  to  get  a  nearer  stretch-out  of  the  circular  line.  Draw  the  centre 
line  of  level  rail  5g"  above  the  floor.  To  fix  heights  and  inclination  of  tangents,  try  a  straight- 
edge in  determining  W  and  B,  points  on  the  upper  and  lower  chord-lines  ;  the  points  W  and  B 
are  not  fixed,  but  may  be  raised  or  lowered  along  the  chord-lines  at  pleasure,  taking  notice,  how- 
ever, that  if  B  is  raised  it  will  increase  the  length  of  ramp  ;  also  if  W  is  raised  the  line  R  0  will 
be  shortened,  a  reasonable  length  of  which  is  required  to  form  the  level  easing  as  shown.  At  B 
draw  B  Y  at  right  angles  to  the  chord  ;  set  off  four  times  5I",  the  lengths  of  the  tangents 
B  D,  D  C,  C  E  and  E  A  of  Fig.  i.  Through  each  of  these  points  of  division  draw  lines  parallel 
to  the  rises  ;  from  W  parallel  to  the  line  of  floor  draw  W  R  ;  connect  R  B  and  prolong  to  0, 
and  B  to  F  indefinitely.  Where  the  line  R  B  cuts  the  vertical  lines  at  T,  J  and  L,  draw  the 
lines  T  C,  J  E  and  L  A  at  right  angles  to  the  rise-lines.  Let  B  F  at  the  ramp  and  the  same 
distance  R  E  at  the  level  easement  be  the  allowance  for  straight  wood  to  be  left  on  those 
ends  of  the  wreath.  At  Fig.  i  make  D  T,  C  J,  E  L  and  A  R  each  equal  D  T  of  Fig.  2.  Con- 
nect T  B,  J  D,  L  C  and  R  E. 

To  Prepare  for  and  Develop  the  Centre  Line  of  Wreath. — At  Fig.  i  draw  G  N  and  S  P 
parallel  to  F  E  ;  parallel  to  A  B  draw  2  H  and  P  Q  ;  parallel  to  C  J  draw  Z  K  and  i^i  M.  At 
Fig.  2  place  the  centres  of  balusters  5,  4  U  G  S  on  the  treads  as  at  the  plan,  and  tbroi:!gh 
eacli  of  these  centres  draw  lines  parallel  to  the  rise-lines  indefinitely  ;  at  baluster  5  mr;ke 
2  H  equal  2  H  of  Fig.  i  ;  at  baluster  4  make  8,  6  equal  D  T  of  Fig.  i  ;  at  balurter  U  make 
Z  K  equal  Z  K  of  Fig.  i  ;  at  baluster  G  make  N  M  equal  N  M  of  Fig.  i  ;  at  bj luster  S  make 
P  Q  equal  P  Q  of  Fig.  i.  Through  B  H  6  K  M  Q  W  trace  the  centre  line  of  v.-reath.  The 
dotted  lines  are  the  top  and  bottom  of  the  wreath  set  off  from  the  centre  li::e. 

To  Find  the  Lengths  of  Balusters  Around  the  Wreath  :— Take  for  example  the  baluster 
at  U.    U  V  is  which  added  to  the  usual    height  of    baluster  at  X,  2'. 2",  makes  the  height 

of  this  baluster  on  its  centre  line  from  the  top  of  step  to  under  side  of  wreath  2'.3tV". 

Fig.  3.  Plan  of  Hand-rail  from  the  Quarter-circle  B  C  of  Fig.  i  with  the  Tangents  B  D 
and  D  C. — Let  the  angles  D  T  B  and  C  J  D  equal  the  same  at  Fig.  i  ;  connect  F  D  ;  parallel 
to  F  D  draw  N  R  ;  parallel  to  C  J  draw  V  L  ;  through  B  and  C  draw  B  P  indefinitely  ;  on  D 
as  centre  with   D  J   as  radius  describe  the  arc  J  P. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  Both  Joints  :— Prolong 
DC  to  Y  ;  make  C  Y  equal  C  W  ;  connect  Y  F:  then  the  bevel  at  Y  will  give  the  angle  required. 

Fig.  4.  Face-mould  from  Plan  Fig.  3;  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints.— Let  Z  P,  Z  P  each  equal  Z  P  of  Fig.  3  ;  make  Z  D  at  right  angles 
to  Z  P  and  equal  to  Z  D  of  Fig.  3  ;  connect  D  P  and  D  P  ;  make  D  L,  D  L  each  equal  D  L 
of  Fig.  3.  Through  L  and  L  draw  N  R,  N  R  parallel  to  ZD;  make  L  R  and  L  N  each  equal 
V  R  and  V  N  ;  make  Z  0  and  Z  S  equal  Z  0  and  Z  S  of  Fig.  3  ;  through  P  and  P  draw  N  H, 
N  H  ;  make  P  H  and  P  H  each  equal  P  N  ;  prolong  DP  to  F  ;  make  P  F  equal  B  F  or  RE 
of  Fig.  2.  The  joints  F  and  P  are  made  at  right  angles  to  the  tangents.  Make  N  A  and  N  A 
each  parallel  to  the  tangents.  Through  H  R  D  R  H  of  the  convex  and  N  S  N  of  the  concave 
trace  the  edges  of  the  face-mould.  The  bevel  at  Y  and  Y,  used  to  square  the  wreath-piece 
at  both  joints  as  shown,  is  taken  from  Y  of  Fig.  3.  One  face-mould  (Fig.  4)  in  this  case 
answers  for  both  wreath-pieces.  The  joint  F  joins  the  ramp  at  F,  and  the  same  joint  joins 
E,  the  level  easement  at  the  top.  This  face-mould  is  explained  in  detail  at  Plate  No.  ii. 
The  development  of  the  centre  line  of  wreath  is    explained  in  detail  at    Plate  No.  20,  Figs. 

and  2,  by  repeating  the  first  quarter  A  C,  Fig.  i,  and  the  developnaent  A  E  of  FiG.  2  of  that 
f late.    Sliding  face-moulds  and  squaring  wreath-pieces  are  given  at  Plate  No.  56. 


PLATE  29 


Fig.  I.  Plan  of  the  Bottom  or  Starting  Portion  of  a  Winding  Staircase  making 
a  Half-turn  with  a  lo"  Cylinder  as  given  at  Fig.  2,  Plate  No.  6. — As  this  case  is  the  same 
as  the  one  already  given  at  Plate  No.  28  with  the  exception  of  position,  that  being  the  top  and 
this  the  bottom  of  a  flight  exactly  alike  in  plan,  there  seems  to  be  no  need  of  repeating 
what  has  just  been  given  in  minute  detail;  but  the  plan,  the  elevation,  etc.,  serve  a  most  useful 
purpose  in  showing  at  a  glance  such  differences  as  are  naturally  caused  by  the  changed  position 
of  the  same  plan:  such  as  the  length  of  ramp  (which  at  the  bottom  of  a  flight  is  shorter); 
also  the  lengths  of  balusters  if  required.  But  there  need  be  no  change  of  face-mould  if  care 
is  taken  in  keeping  the  inclination  of  tangents  alike.  It  will  be  seen  upon  examination  that 
all  measurements  required  at  Fig.  2  are  taken  from  Fig.  i,  or  the  reverse  ;  those  measuiements 
taken  from  Fig.  2  as  required  at  Fig.  i  are  lettered  or  figured  alike.  The  lettering  likewise 
agrees  between  the  quarter  B  C  of  Fig.  i  and  the  plan  of  hand-rail  Fig.  3.  Also  as  far  as 
possible  the  lettering  is  alike  between  the  plan  of  hand-rail  FiG.  3  and  the  measurements  to 
be  taken  in  connection  with  it  for  drawing  the  face-mould  and  squaring  the  wreath-piece  at 
Fig.  4. 


Plate  No.  29 


Plate  No. 30 


Scale  IVa  1n.=  1  Ft. 


PLATE  30. 


Staircases  are  frequently  planned  of  greater  width  at  the  starting  by  curving  the  front- 
string  out,  embracing  in  the  curve  from  one  to  five  treads;  the  least  curve  including  but  one 
step  is  done  merely  to  save  the  width  of  stairs  at  this  point  by  setting  the  newel-post  a  few 
inches  aside.  The  larger  curves  including  more  steps  give  the  stairs  an  inviting  and  more 
ornamental  appearance.  There  is  also  a  more  recent  practice  of  setting  the  newel  on  top  of 
the  first  step  by  extending  this  step  and  riser  in  a  curve  sufficient  to  include  the  curve-out 
and  the  base  of  the  newel  as  shown  at  Fig.  3.  This  gives  in  very  little  space  a  neat  and 
elegant  finish  at  the  starting  of  a  staircase. 

Fig.  I.  Plan  of  Curve-out  in  One  Tread,  together  with  Sufficient  Elevation  for 
those  Measurements  Required  to  Fix  the  Height  of  Newel  and  Draw  the  Face-mould 
or  Parallel  Pattern.— Let  the  bottom  of  rail  rest  on  the  centres  of  the  short  balusters  at  XX; 
make  R  T  equal  6",  and  T  L  half  the  thickness  of  rail;  let  L  N  be  the  centre  line  of  rail; 
draw  L  M  at  right  angles  to  the  rise-line;  parallel  to  the  rise-line  draw  L  A:  then  A  at  the 
plan  on  the  centre-line  of  rail  C  A  becomes  a  fixed  point  from  which  the  tangent  A  B  may 
be  drawn  at  pleasure.  Make  C  F  equal  M  N;  connect  FA;  from  any  point  on  the  centre  curve 
S  draw  a  line  parallel  to  A  B  touching  the  line  C  A  at  Z;  draw  Z  0  parallel  to  C  F;  at  right 
angles  to  A  B  draw  C  D  indefinitely;  on  A  as  centre  with  A  F  as  radius  describe  the  arc  F  D; 
connect  D  B.     See  Plate  No.  74,  Figs.  3  and  4. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  B: — 
Parallel  to  A  B  draw  C  K  indefinitely;  at  right  angles  to  AB  draw  B  J;  make  J  K  equal  C  F; 
connect  K  B:  then  the  bevel  at  K  will  give  the  angle  sought. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  C: — 
Prolong  the  tangent  B  A  to  G,  and  the  tangent  A  C  to  H,  indefinitely;  make  C  H  equal  C  E; 
connect  H  G:  then  the  bevel  at  H  will  give  the  angle  required. 

Fig  2.  Parallel  Pattern  for  Wreath-piece  from  the  Plan  Fig.  i. — Make  AO  VF  equal 
the  same  letters  at  Fig.  i.  On  F  as  centre  with  D  B  of  Fig.  i  as  radius  describe  an  arc  at  B; 
on  A  as  centre  with  A  B  of  Fig.  i  as  radius  intersect  the  arc  at  B;  connect  B  A;  make  B  T 
equal  B  T  of  Fig.  i;  through  T  at  right  angles  to  A  B  draw  Y  Z;  draw  0  S  parallel  to  A  B; 
make  0  S  equal  Z  S  of  Fig.  i.  The  width  of  the  rail  being  3",  the  parallel  pattern  will  be  3!". 
Make  VP  and  VQ  each  if;  parallel  to  V  F  draw  P  G  and  Q  E;  make  B  L  and  B  W  each  ij"; 
parallel  to  B  A  draw  L  Z  and  W  Y;  on  S  as  centre  with  ij"  as  radius  describe  a  circle  and 
sketch  the  curves  P  Z  and  Q  Y  touching  the  circle.  The  angle  for  squaring  the  wreath-piece  at 
joint  B  is  taken  by  the  bevel  K  at  Fig.  i,  and  for  the  joint  F  the  bevel  H  of  Fig.  i. 

Fig.  3.  Plan  and  Elevation  of  the  Starting  of  a  Staircase  with  the  Front-string 
Curved  Out  and  the  Newel  Set  on  Top  of  the  First  Step ;  the  Elevation  and  Plan 
Prepared,  Fixing  the  Height  of  Hand-rail  at  the  Newel,  and  for  Drawing  the  Face- 
mould. — Let  the  bottom  of  rail  rest  on  X  X,  the  centres  of  short  balusters,  and  make  A  B,  the 
centre  line  of  rail,  parallel  to  X  X;  make  D  E  equal  8",  and  E  F  half  the  thickness  of  rail;  draw 
F  C  parallel  to  the  line  of  tread;  draw  A  G  parallel  to  the  line  of  riser:  then  G  becomes  a 
fixed  point  on  the  line  of  tangent  V  G  from  which  the  level  tangent  G  J  Z  must  be  drawn, 
but  may  be  kept  any  distance  from  I  to  Z  at  pleasure.  Parallel  to  Z  G  draw  H  0  and  L  K; 
parallel  to  V  M  draw  0  S  and  K  W;  from  V  at  right  angles  to  G  J  draw  V  N  indefinitely; 
on  G  as  centre  with  G  M   as  radius  describe  the  arc   M  N;  connect  N  J. 

To  Find  the  Angle  with  v/hich  to  Square  the  Wreath-piece  at  the  Joint  over  J: — 
From  V  parallel  to  G  Z  draw  V  R  indefinitely;  draw  J  T  at  right  angles  to  J  G;  make  T  R 
equal  V  M;   connect  R  J:    then  the  bevel  at   R  will  give  the  angle  sought. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  V: — 
Prolong  Z  G  to  U,  and  G  V  to  P,  indefinitely;  make  V  P  equal  V  Q;  connect  P  U:  then  the 
bevel  at  P  will  give  the  angle  required. 

Fig.  4.  Parallel  Pattern  for  Wreath-piece  from  the  Plan  Fig  3  — As  this  rail  is  to  be  4" 
wide  by  2V  thick,  the  pattern  will  answer  to  get  out  the  wreath-piece  if  4J"  wide.  Draw  the  line 
GA;  make  G  M  equal  G  M  of  Fig.  3.  Let  M  A  equal  5"  more  or  less  for  straight  wood;  on  M  as 
centre  with  N  J  of  Fig.  3  as  radius  describe  an  arc  at  J ;  on  G  as  centre  with  G  J  of  Fig.  3 
as  radius  intersect  the  arc  at  J;  connect  J  G;  make  G  SW  equal  G  S  W  of  Fig.  3;  parallel 
to  J  G  draw  S  H  and  W  L;  make  W  L  equal  K  L  of  Fig.  3;  make  S  H  equal  0  H  of  Fig.  3. 
With  2|"  radius  describe  circles  from  the  centres  J,  H,  L,  M  and  A;  connect  BF  and  C  E.  The 
joints  J  and  A  are  made  at  right  angles  to  the  tangents.  Bend  a  flexible  strip  of  wood  touching 
the  circles  on  the  convex  and  concave  and  mark  the  curved  edges  of  the  pattern.  The  angle  for 
squaring  the  wreath-piece  at  joint  J  is  taken  by  the  bevel  at  R  of  Fig.  3,  and  for  the  joint  A  by  the 
bevel  at  P  of  Fig.  3.  The  height  of  hand-rail  from  the  top  of  the  second  step,  D,  to  the  bottom 
of  the  rail,  E,  will  equal — by  adding  the  height  of  short  baluster  at  X,  which  is  2'. 2",  to  the  8" 
raised  between  D  and  E — 2'.  10".  Face-mould  for  Figs,  i  or  3  is  treated  in  detail  by  Plate  No.  13. 
One  feature  of  this  plan  given  at  Fig.  3  which  is  open  to  objection  is  the  increased  heigJit  of  newels  j  how 
to  reduce  this,  if  required,  will  be  shoioti  by  the  following  Plate,  No.  31. 


PLATE  31. 


Fig.  I.  Plan  and  Elevation  of  the  Starting-  of  a  Staircase  with  the  l^ront-string 
Curved  Out  and  the  Newel  Set  on  Top  of  the  First  Step,  the  Elevation  and  Plan  Pre- 
pared to  Continue  the  Hand-rail  on  a  Common  Inclination  to  the  Newel,  and  for  a  Face- 
mould  as  Required. — Let  the  l)ottom  of  tlie  rail  rest  on  X  X  the  centres  of  sliort  balusters  ; 
draw  the  centre  line  of  rail  Z  R  parallel  to  X  X;  make  the  tangents  C  V  and  V  P  of  equal 
lengths;  from  V  parallel    to  the  rise-line  draw  V  5;  parallel  to    the    tread-line  draw  5  Y;  make 

4  R  equal  V  P;  through  R  parallel  to  V  4  draw  W  8;  at  R  draw  R  4  at  right  angles  to  V  4; 
draw  R  U  at  right  angles  to  R  5;  make  C  G  equal  Y  Z;  connect  G  V;  make  V  B  equal  4,  5  and 
at  right  angles  to  V  P;  connect  B  P;  through  C  draw  P  D  indefinitely;  on  V  as  centre  with 
V  G  as  radius  describe  the  arc  G  D;  at  right  angles  to  D  P,  through  V  draw  Q  0,  F  N  M  and 
T  L;  at  riglit  angles  to  C  V  draw  A  K  and   L  J. 

To  Find  the  Angle  with  which  to  Square  Both  Ends  of  the  Wreath-piece : — Parallel  to 
C  V  draw  K  I;  at  right  angles  to  G  V  draw  I  H;  prolong  C  V  to  E  indefinitely;  make  C  E 
equal  I  H  ;  connect   E  F,  then  the  bevel  at  E  will  give  the  angle  required. 

Fig.  2.  Face-mould  from  Plan,  Fig.  i;  also  Squaring  the  Wreath-piece  at  Both  Joints. 
— Draw  the  line  D  D;  make  S  D,  S  D  each  equal  S  D  of  Fig.   i;  at  right  angles  to  D  D  through 

5  draw  Q  0;  make  S  V  equal  S  V  of  Fig.  i;  connect  V  D  and  V  D;  make  V  K  J  and  V  K  J 
each  equal  V  K  J  of  Fig.  i.  At  right  angles  to  D  D  through  J  K  and  J  K  draw  T  J,  N  M, 
N  M  and  T  J;  make  J  T  and  J  T  each  equal  L  T  of  Fig.  i;  make  K  M,  K  N  and  K  M,  K  N  each 
equal  A  M  and  A  N  of  Fig.  i;  make  S  0  and  S  Q  equal  S  0  and  S  Q  of  Fig.  i;  through  D 
draw  T  Z  at  both  ends;  make  D  Z  and  D  Z  each  equal  T  D.  The  lower  end  of  the  wreath- 
piece  requires  an  addition  of  straight  wood  to  fill  out  the  plumb  joint  as  shown  at  6  U  of 
the  elevation,  Fig.  i.  Make  D  F  equal  6  U  of  Fig.  i;  make  D  U  equal  4",  more  or  less  for 
straight  wood;  parallel  to  V  U  draw  Z  B  and  T  A.  The  joints  U  and  F  are  made  at  right 
angles  to  the  tangents.  Through  Z  M  0  M  Z  of  the  convex  and  T  N  Q  N  T  of  the  concave 
trace  the  curved  edges  of  the  face-mould.  The  angle  for  squaring  the  wreath-piece  at  joints 
F  and  U  is  taken  by  the  bevel  at  E,  Fig.  i.  This  case  of  face-mould  is  treated  in  detail  at 
Plate  No.  15.  The  difference  in  height  between  this  manner  of  treating  the  hand-rail  and 
that  of  Fig.  3,  Plate  30,  is  from  the  top  of  the  rail  at  8,  where  it  joins  the  newel  in  that 
case,  to  the  top  of  the  rail  at  2,  where  it  joins  the  newel  in  this  case,  just  6".  When  the  rail 
is  set  up  at  X  X,  2'. 2",  the  height  of  rail  from  W  to  2  will  be  2'. 6^",  and  between  those 
points  at  Fig.  3,  Plate  30,  the  height  is  3'..j".  Much  more  than  6"  can  be  gained  if  desired 
by  moving  the  centre  of  the  newel  further  towards  the  first  riser  on  the  line  3,  3  and  increas- 
ing the  radius  of  the  curve-out  to  suit  the  change.  This  might  be  very  desirable  in  case  of 
designing  with  this  style  of  finish  a  low  newel. 

Fig.  3.  Plan  of  a  Quarter-platform  Stairs  with  a  Quarter-cylinder  of  8"  Radius,  the 
Rise  rs  each  Set  at  A  and  B,  the  Chord-lines. — Let  A  Z  B  be  the  centre  line  of  rail  on  the 
plan;  divide  the  quarter-circle  A  B  at  Z  in  exactly  two  equal  parts  ;  connect  6  Z  ;  at  right  angles 
to  6  B  draw  B  E  indefinitely;  through  A  at  right  angles  to  6  A  draw  THY  indefinitely;  through 
Z  at  right  angles  to  6  Z  draw  H  E;  make  A  Y  equal  9",  one  tread;  at  right  angles  to  A  Y 
draw  Y  P;  make  YP  equal  8" — the  rise;  connect  PA;  parallel  to  Y  P  draw  H  K;  parallel 
to  AY  draw  K  L;  make  L  M  equal  Y  L;  divide  P  M  in  two  equal  parts  at  N;  make  H  J 
equal  M  N;  draw  J  W  parallel  to  Y  A;  draw  WX  parallel  to  5  A;  from  A  draw  A  R  parallel 
to  the  tangent  H  Z;  make  A  R  equal  the  tangent  H  Z;  connect  X  R,  w^hich  is  the  level 
line.  Through  H  draw  5  G  parallel  to  X  R;  anywhere  on  the  centre  curve-line  draw  the  line  2  U 
parallel  to  X  R;  from  A  at    right  angles  to  X  R  draw  A  Q    indefinitely;  on  the  dividing  radial 

6  D  make  Z  D  equal  M  N;  connect  D  H;  from  Z  at  right  angles  to  H  G  draw  Z  I  indefinitely; 
on  H  as  centre  with  H  D  as  radius  describe  an  arc  cutting  the  line  Z  I  at  C;  on  H  again 
as  centre  with   K  A  as  radius  describe  an  arc  cutting  the  line  A  Q  at  Q;  connect  Q  C. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  Z  : — 
Make  Z  F  equal  Z  0;  connect  F  G;  then  the  bevel  at  F  will  give  the  angle  sought. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  A : 
— Make  A  T  equal   H  V;  connect  T  5;  then  the  bevel  at  T  will  give  the  angle  required. 

Fig.  4.  Parallel  Pattern  for  Wreath-piece  from  the  Plan  Fig.  3. — Draw  the  line  Q  C, 
and  make  Q  4  C  equal  the  same  at  Fig.  3.  On  Q  as  centre  with  A  K  of  Fig.  3  describe  an 
arc  at  K;  on  C  as  centre,  with  D  H  of  Fig.  3  as  radius,  intersect  the  arc  at  K;  also  4  K 
will  equal  4  H  of  Fig.  3.  Connect  Q  K,  K  C  and  4  K;  make  K  3  equal  H  3  of  Fig.  3;  make 
Q  V  equal  A  V  of  Fig.  3;  draw  V  2  parallel  to  K  4;  make  V  2  equal  U  T  of  Fig.  3;  make  Q  B 
equal  3",  or  more  for  straight  wood.  The  size  of  this  rail  is  to  be  2^'  thick  by  3"  wide, 
therefore  the  parallel  pattern  will  require  to  be  3!"  wide.  With  \y  as  radius  on  C,  3,  2, 
Q,  B  as  centres,  describe  circles.  Make  a  line  touch  these  circles  for  the  concave  and  convex 
edges  of  the  pattern.  The  joints  B  and  C  are  made  at  right  angles  to  the  tangents.  The 
angle  for  squaring  the  wreath-piece  at  joint  C  is  taken  by  the  bevel  at  F  of  Fig.  3,  and  for 
the  joint  B  by  the  bevel  at  T  of  Fig.  3.  The  slide  line  is  drawn  at  right  angles  to  the 
level  line  4  K.  This  pattern  serves  for  both  wreath-pieces,  C  being  the  centre  joint.  A  face- 
mould  of  this  kind  is  treated  in  detail  at  Plate  No.  16.  This  quarter-cylinder  with  its  con- 
necting steps  and  risers  should  be  planned  in  such  a  way  that  the  quarter-wreath  could  be 
got  out  in  one  piece  of  a  common  inclination  as  shown  at  Plate  No.  37,  Figs.  5,  6  and  7. 
It  would  cost  less  and  be  a  superior-shaped  wreath-piece.    See  Plate  No.  78. 


Plate  No.  31 


PLATE  32. 


Fig.  I.  Plan  of  Hand-rail  composed  of  Two  Curves  of  Different  Radius  as  a  Curve- 
out  at  the  Starting  of  a  Staircase,  taken  from  the  Plan  given  at  Fig.  5,  Plate  No.  6.— 

This  shape  of  curve  is  a  necessity  in  order  to  make  a  proper  connection  with  a  square  newel 
where  the  sides  of  the  newel  are  required  to  be  set  parallel  to  the  sides  of  tiie  hallway,  as 
siiown  by  the  plan  above  mentioned.  An  elevation  has  to  be  set  up  in  order  to  fix  the  length 
of  plan  tangent  D  F,  as  at  A  C  of  Fig.  2.  And  the  greater  the  required  height  of  newel  tiie 
higher  up  the  line  A  C  must  be  placed,  and  therefore  the  shorter  this  line  and  the  plan  tangent 
will  be. 

Fig.  2.  Elevation  of  Tread  and  Rise  as  at  Plan  of  Hand-rail  Fig.  i,  taken  from 
Fig,  5  of  Plate  No.  6. — Let  the  bottom  line  of  rail  pass  through  X  X,  the  centres  of  the  short 
balusters  of  the  regular  tread;  make  LQ  5^^",  more  or  less,  depending  on  what  height  of  newel 
is  demanded;  make  QV  half  the  thickness  of  rail;  draw  VGA  parallel  to  the  line  of  treads; 
let  C  B  be  the  centre  line  of  rail  parallel  to  XX.  When  the  hand-rail  is  set  up,  the  height  from 
the  top  of  the  first  step,  L,  to  the  bottom  of  the  rail,  Q,  will  be  2'. 2"  at  X  and  5^"  more  at  L  Q, 
equal  to  2'.'jV.  At  Fig.  i  maice  D  E  equal  A  B  of  Fig.  2;  connect  E  F;  parallel  to  D  E  draw  H  T 
I  U,  J  V,  K  W  and  LY. 

Fig.  3.  Face-mould  to  be  taken  from  Plan  Fig.  i ;  also  Squaring  the  Wreath-piece 
at  Both  Joints. — Make  FTU  VWYE  equal  the  same  at  Fig.  i;  draw  FG  at  right  angles  to  FE; 
make  FG  equal  FG  of  Fig.  i;  tiirough  G  parallel  to  FE  draw  AH;  parallel  to  FG  draw  T  H, 
U  I,  VJ,  W  K,  0  L  and  N  M;  make  G  A  equal  G  H.  Set  off  all  measurements  from  the  line  F  E  as 
taken  on  the  line  F  D  of  Fig.  i  according  to  the  corresponding  letters  at  the  curves.  Through 
NOXXXXXA  of  the  convex  and  M  LKJ  IH  of  the  concave  trace  the  curved  edges  of  the  face- 
mould.  The  bevel  at  E  of  Fig.  i  is  used  to  square  the  wreath-joint  G  of  Fig.  3.  It  would  be 
well  to  add  about  3"  for  straight  wood  to  joint  E. 

Fig.  4.  Plan  of  the  Starting  Portion  of  a  Staircase  with  the  Front-string  Curved 
Out,  and  Embracing  Four  Treads  of  Equal  Widths  at  the  Wall-string  and  Front- 
string. — At  the  elevation.  Fig.  5,  the  bottom  line  of  rail  rests  at  X  X,  the  centres  of  short 
balusters;  BC  and  DC  are  the  centre  lines  of  rail.  Fixing  the  point  C  controls  the  height 
of  rail  at  the  newel;  also  the  length  of  tangent  A  C  at  Fig.  4.  FE  being  9",  add  that  to  the 
height  of  short  baluster  at  X,  2'.2",  and  the  sum  2'. 11"  will  be  the  height  between  F  and  E. 
In  a  flat  curve  like  this  it  is  desirable  to  keep  the  point  C  up  as  high  as  can  be  allowed,  for 
it  shortens  the  tangent  A  C,  Fig.  4,  and  makes  the  level  line  C  0  rriore  nearly  a  tangent.  Let 
the  angle  ABC  equal  the  same  at  the  elevation;  connect  CO;  parallel  to  CO  draw  4  W  I  3 
QU,  RG,  8K  and  AM;  parallel  to  A  B  draw  4  H,  S  V,  U  Z,  3Y  and  8  X.  From  A  draw  A9  at 
right  angles  to  0  C;    on  C  as  centre  with  C  B  as  radius  desci  ibe  tiie  arc   B9;  connect  9  0. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  over  the  Joint  A:— 
Draw  S6  parallel  to  CB;  prolong  CA  to  L;  make  AL  equal  A6;  connect  LG:  then  the  bevel 
at  L  will  give  the  angle  sought. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  over  the  Joint  0:— 
Draw  0  2  at  right  angles  to  OC;  make  2  M  equal  A  B;  connect  M  0:  then  the  bevel  at  M  will  give 
Lue  angle  sought. 

Fig.  6.  Face-mould  from  Plan  Fig.  4 ;  also  Squaring  the  Wreath-piece  at  Both 
Joints. — Let  OB  equal  0  9  of  Fig.  4.  On  B  as  centre  with  B  C  of  Fig.  4  as  radius  descril)e 
an  arc  at  C;  on  0  with  0  C  of  Fig.  4  as  radius  describe  an  intersecting  arc  at  C;  connect  O  C 
and  C  B;  make  the  spaces  lettered  on  B  C  agree  with  those  of  B  C  at  Fig.  4;  parallel  to  0  C 
draw  XJ,  Y  I,  Z  Q,  VTR  and  H  W:  take  all  measurements  on  the  tangent  AC,  Fig.  4,  and  set 
them  off  from  the  line  B  C  as  shown  by  the  corresponding  letters;  make  ON  equal  OJ;  through 
B  draw  WK;  make  BK  equal  BW;  prolong  C  B  to  A;  make  BA  4"  for  straigiit  wood.  The 
joints  A  and  0  are  at  right  angles  to  the  tangents.  Through  WT5FEPN  of  the  convex  and 
KRQIJ  of  the  concave  trace  the  curved  edges  of  the  face-mouUl.  The  angle  for  squaring  the 
wreath-piece  at  joint  0  is  taken  by  the  bevel  M  at  Fig.  4,  and  that  for  joint  A  by  the  bevel  L 
at  Fig.  4.  Face-mould  Fig.  3  is  treated  in  detail  at  Plate  No.  10,  and  face-mould  Fig.  6  is 
treated  likewise  at  Plate  No.  13. 


Note. — The  line  O  C  of  Figs.  4  and  6  is  not  a  tangent,  but  is  simply  a  level  line  used  to  fix  the  height,  control  the 
joint,  and  to  guide  the  measure,  or  drawing,  of  the  face-mould.  Care  should  be  taken  in  laying  out  the  squaritii;  at  joint  O 
of  the  7ureath-pieee  that  the  width  of  rail  he  made  equal  /<?  K  N  of  FiG.  4,  because,  although  1^  N  is  at  right  angles  to  O  C,  it 
is  oblique  to  the  curve,  and  may,  therefore,  measure  a  quarter  of  an  inch  or  more  over  the  width  of  rail.  After  the  wreath- 
piece  is  squared  at  the  joint  O,  a  portion  of  the  wood,  equal  to  N  U  K  of  FiG.  4,  is  cut  away  to  correct  that  joint. 


PLATE  33. 


Fig,  I.  Plan  of  the  Landing  Portion  of  a  Straight  Flight  of  Stairs  with  the 
Top  Riser  Set  in  the  Whole  Depth  of  a  Ten-inch  Cylinder,  thereby  Saving  Five 
Inches  Space. — Tlie  plan  made  as  described  with  the  addition  of  the  plan  and  centre 
line  of  rail,  proceed  to  set  up  the  elevation,  Fig.  2,  resting  the  bottom  line  of  rail  on 
the  centres  XX  of  the  short  balusters,  and  the  centre  of  rail  A  D  in  position  parallel  to 
X  X,  the  point  D  being  fixed  by  the  level  line  G  D,  and  its  height  F  G  from  the  floor. 
The  point  A  is  decided  by  the  position  of  the  chord-line  as  taken  from  the  plan  ;  A  C  is 
drawn  parallel  to  the  line  of  tread.  This  completes  the  preparation  of  the  elevation,  all  being 
obtained  that  is  required  in  fixing  the  point  A  and  D  together  with  the  height  C  D.  Again 
at  Fig.  i,  parallel  to  the  line  of  string,  let  the  line  F  A  pass  through  C,  the  centre  of  rail  ; 
make  C  D  at  right  angles  to  C  A,  and  C  D  A  equal  to  C  D  A  of  Fig.  2.  From  A  draw  the 
line  A  I  touching  the  centre  line  of  rail  at  Q  ;  the  exact  place  of  Q  is  determined  by  draw- 
ing a  line  from  the  centre  0,  at  right  angles  to  A  Q;  parallel  to  A  Q  draw  R  Y,  S  X,  V  W  H  J 
and  P  U;  parallel  to  C  D  draw  T  2,  N  M,  X  3  and  YZ;  from  C  at  right  angles  to  A  Q  draw 
C  E;  on  A  as  centre  describe  the  arc  D  E;  connect  E  Q. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  Q  : — 
Anywhere  along  the  line  W  J  draw  H  I  parallel  to  0  Q.  Make  H  J  equal  N  M;  connect  J  I; 
then  the  bevel  at  J   will  give  the  angle  sought. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  C:— 
From  M  parallel  to  A  C  draw  M  K;  make  C  F  equal  K  L;  connect  F  G;  then  the  bevel  at  F 
will  give  the  angle  required. 

Fig.  3.  Face-mould  from  Plan  Fig.  i,  also  showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints. — Draw  the  line  D  Q  indefinitely  ;  make  D  Q  equal  E  Q  of  Fig.  i. 
On  D  as  centre,  with  D  A  of  Fig.  i  as  radius,  describe  an  arc  at  A;  and  on  Q  as  centre,  with 
Q  A  of  Fig.  i  as  radius,  intersect  the  arc  at  A.  Connect  A  Q  and  A  D;  prolong  A  D  to  L  and 
equal  to  A  E  of  Fig.  2,  or  at  pleasure  for  straight  wood.  Make  D  2  M  3  Z  the  same  as  at 
the  corresponding  letters  of  Fig.  i.  Parallel  to  A  Q  draw  Z  R,  3  S,  M  V  W  and  2  U  P;  through 
Q  draw  R  C  at  right  angles  to  A  Q  ;  make  Q  C  equal  Q  R;  make  A  4  equal  A  4  of  Fig.  i  ; 
make  3,  5  S  equal  X  5  S  of  Fig.  i;  make  M  V,  M  W  equal  N  V,  N  W  of  Fig.  i;  make  2  U,  2  P 
equal  T  U,T  P  of  Fig.  i;  through  D  draw  P  B;  make  D  B  equal  D  P;  parallel  to  D  L  draw 
P  E  and  B  F;  through  C  4  5  V  U  B  of  the  convex  and  R  S  W  P  of  the  concave  trace  the  edges 
of  the  face-mould.  The  tangent  Q  A  being  a  level  line,  the  face-mould  slides  along  the  joint 
R  C.  The  angle  with  which  to  square  the  wreath-piece  at  joint  Q  is  taken  by  the  bevel  at  J  of 
Fig.  I,  and  for  the  joint  L  is  taken  by  the  bevel  at  F  of  Fig.  i.  This  face-mould  is  treated  in 
detail  at  Plate  No.  14.  The  level  poriiun  of  rail  Q  B,  Fig.  i,  may  have  as  much  straight  wood 
attached  to  B  as  seems  desirable. 

Fig.  4.  Plan  of  the  Starting  Portion  of  the  Same  Flight  of  Stairs  given  at 
Fig.  I  with  a  Ten-inch  Cylinder,  and  the  Starting  Riser  Set  in  the  Whole 
Radius  or  Depth  of  Cylinder,  Saving  Another  Five  Inches. — Having  made  the  plan 
as  described,  proceed  to  set  up  the  elevation.  Fig.  5.  Let  the  bottom  line  of  rail  rest 
on  the  centres  X  X  of  the  short  balusters.  Place  the  centre  of  the  rail  C  B  in  position  par- 
allel to  X  X,  the  point  C  being  fixed  by  the  level  line  G  C,  and  its  height  F  G  from  the 
floor.  The  point  B  is  fixed  by  the  position  of  the  chord  line  as  taken  from  the  plan.  A  C  G 
is  drawn  parallel  to  the  line  of  floor.  The  place  of  B,  the  height  A  B,  and  the  distance  A  C 
is  all  that  is  required  of  the  elevation.  Again  at  Fig.  4  prolong  the  diameter-line  S  A  of  the 
cylinder  to  B  indefinitely;  at  the  centre  line  of  the  rail  A,  at  right  angles  to  A  B,  draw  A  C; 
make  ABC  equal  to  A  B  C  of  Fig.  5.  From  C,  Fig.  4,  draw  the  line  C  D  tangent  to  the  centre  line 
of  the  rail  at  the  point  D,  to  be  determined  by  drawing  a  line  from  0  at  right  angles  to  the 
tangent  C  D.  The  remaining  portion  of  the  rail  D  S  is  level,  and  at  S  straight  wood  can  be 
added  at  pleasure.  To  draw  the  face-mould  proceed  as  at  Figs,  i  and  3.  It  is  necessary  to 
give  further  attention  to  this  case  of  hand-rail  from  the  fact  that  although  the  riser  is  set 
in  the  cylinder  the  same  at  the  bottom  as  at  the  top  of  the  flight,  there  is  a  difference 
requiring  another  face-mould  unless  it  is  thought  worth  while  to  make  the  top  face-mould  answer 
also  for  the  bottom.  In  the  latter  case  let  C  stand  exectly  as  it  is,  and  make  C  L  equal 
C  A  of  Fig.  i  and  L  N  equal  C  D  of  Fig.  i;  then  change  the  place  of  the  chord-line — or 
commencement  of  the  cylinder — to  L,  and  describe  the  centre  line  of  the  rail  from  the  new 
centre,  and  draw  from  C  a  new  tangent,  and  the  case  will  then  agree  with  the  top  plan 
tangents  and  face-mould. 

Fig.  6.  Elevation  Same  as  Fig.  5,  for  the  Development  of  the  Centre  Line  of  Wreath 
from  Plan,  Fig.  4. — The  corresponding  letters  and  figures  of  Figs.  4  and  6  will  give  a 
snfiicient  explanation.  This  developm.ent  of  the  centre  line  of  the  wreath-piece  is  introduced 
to  demonstrate  the  correctness  of  treating  such  cases  in  the  manner  here  shown.  Treating 
cases  of  hand-railing  like  this  in  two  quarters — the  joint  in  the  centre — makes  it  necessary  to 
square  up  both  pieces  and  draw  two  face-moulds,  with  no  better  result  and  nearly  double  the 
cost.  This  face-mould.  Fig.  3,  is  treated  in  detail  at  Plate  No.  14.  The  development  of  the 
centre  line  of  wreath  is  given  ip  detail  at  Plate  No.  21,  Figs,  i  and  2. 


Scale  IV2  In.  =  1  Ft. 


Plate  No.  34 


PLATE  34. 

A  Wreath  in  one  Piece  over  a  Twelve-inch  Cylinder. — Size  of  Hand-rail  Four  Inches  Wide  by  Two  and 
A  Half  Inches  Thick;  the  Wreath  to  be  Got  out  of  Plank  Four  Inches  Thick. 

Fig.  I.  Plan  of  the  top  portion  of  a  straight  flight  of  stairs  with  a  12''  cylinder,  the  landing  riser  set  in 
the  whole  depth  of  the  cylinder. 

Fig.  2.  Elevation. — After  drawing  the  plan  as  given  and  described,  set  up  one  tread  and  rise;  rest  the 
bottom  of  tlie  rail  at  X  X,  the  centres  of  the  sliort  balusters.  Tiie  bottom  of  the  level  rail  is  set  up 
from  the  floor  4"  as  usual.  Place  the  chord-line  as  at  plan.  From  E,  the  intersection  of  the  centre 
lines  of  rail  draw  E  P  parallel  with  riser,  and  from  J   draw  J  P  parallel  with  tread. 

Fig.  I.  To  Prepare  the  Plan  for  Drawing  the  Face-mould: — Make  J  P  equal  J  P  of  Fig.  2.  At 
right  angles  to  J  P  draw  P  E  equal  to  P  E  of  Fig.  2.  At  right  angles  to  J  T  draw  T  U  equal  to  P  E  ; 
connect  J  U  and  prolong  the  line  indefinitely  ;  connect  T  P  ;  parallel  to  T  P  touching  the  centre  line  of 
rail  draw  Z  C  ;  from  C  parallel  to  P  E  draw  C  D  ;  from  Z  parallel  to  T  U  draw  Z  7  :  then  the  height 
C  D  touching  the  inclined  line,  J  E  is  equal  to  the  iieight  Z  7  touching  the  inclined  line  J  U  ;  and  both 
inclinations  (whatever  they  may  be)  are  in  the  same  plane.  From  the  centre  L  at  right  angles  to  Z  C 
draw  L  A  ;  parallel  to  Z  C  draw  6  Q,  L  N  and  K  W  ;  from  J  at  right  angles  to  Z  C  draw  J  B  indefinitely  ; 
on  C  as  centre  witii  D  J  as  radius  describe  an  arc  at  B  ;  connect  B  A.  To  Find  the  Angle  ivitli  which  to 
Square  the  Wreath  at  the  Joint  over  J  : — Draw  L  8  at  right  angles  to  6  K  ;  make  L  8  equal  N  H  ;  connect 
8  J:  then  the  level  at  8  will  give  the  angle  required.  To  Find  the  Slide  Distance  {or  JSIovement  of  Face-mould') 
to  Plumb  the  Sides  of  the  Wreath: — From  F  draw  K2  at  right  angles  to  DC;  make  AS  equal  2  D;  connect 
S  5  :  draw  S  4  at  right  angles  to  S  5  and  equal  to  2" — half  the  thickness  of  plank — through  4  parallel 
to  S  5  draw  4  R  :  then  4  R  will  be  the  distance  sought. 

Fig.  3.  Face-mould  to  Include  the  Whole  Cylinder  from  Plan  Fig.  i. — Draw  the  line  J  A  equal 
to  B  A  of  Fig.  i.  On  J  with  J  D  of  Fig.  i  as  radius  describe  an  arc  at  D  ;  on  A  with  A  C  of  Fig.  i 
as  radius  intersect  the  arc  at  D  ;  connect  D  A  and  D  J,  prolonging  eaeh  indefinitely.  Make  J  I  G  F  E  equal 
the  same  at  Fig.  i  ;  parallel  to  D  A  draw  E  T,  F  6,  G  L  and  I  W  K  ;  make  D  X  Y  equal  C  X  Y  of  Fig.  i  ; 
make  E  9  V  T  equal  P  9  V  T  of  Fig.  i  ;  make  F  1 ,  3,  6  equal  Q  1 , 3,  6  of  Fig.  i  ;  make  G  0  M  L  equal  N  0  M  L 
of  Fig.  I,  and  W  I  K  equal  W  9  K  of  Fig.  i  ;  connect  T  J  and  prolong  the  line  both  ways  indefinitely. 
Make  J  1 1  equal  J  K  ;  make  T  1  3  equal  T  6  ;  connect  L  A  ;  make  A  1 0  equal  A  V  ;  make  J  1  2  equal  2"  for 
straight  wood  ;  make  the  joint  12  at  right  angles  to  J  D — the  joint  6,  1 3  is  not  changed — draw  K  8  and 
11,4  parallel  to  J  D.  Through  6  V  3  M  K  of  the  concave  and  13,  Y,  10,  X,  9,  1 0,  W,  1 1  of  the  convex  trace 
the  curved  edges  of  the  face-mould.  Over-wood  has  to  be  added  to  the  wreath  from  joint  T  to  find  this. 
From  T  draw  T  B  parallel  to  10,  V  ;  make  T  B  equal  4  R  of  Fig.  i  ;  then  B  C  will  equal  the  required 
over-wood. 

Fig.  4.  Squaring  the  Wreath. — This  wreath  with  its  joints  is  to  be  sawed  out  square  as  usual, 
and  the  lower  joint  made  square  from  the  face  of  plank.  In  cutting  out  the  wreath,  if  more  wood 
be  allowed  as  indicated  by  the  small  dots,  x  x  ,  it  will  prove  an  advantage  in  squaring  up.  The  angle  for 
squaring  the  wreath  at  its  lower  joint  8  is  taken  by  the  bevel  8  at  Fig.  i.  The  line  T  is  the  joint 
of  face-mould,  and  T  0 — equal  to  B  C  of  Fig.  3 — is  the  wood  added  to  the  wreath,  required  to  make  a 
complete  plumb-joint  through  the  thickness  of  the  plank.  To  Slide  the  Face-mould  in  Position  to  Plirmh  the 
Sides  of  the  Wreath  and  the  Upper  Joint: — Draw  from  T  and  from  1  the  lines  1,  2  and  T  B  parallel  to  V  10; 
make  T  B  and  1,  2  each  equal  4  R  of  Fig.  i  ;  then  place  the  joints  of  the  face-mould  on  their  centres 
at  2  and  B,  marking  joint  B  and  the  point  B  ;  also  the  edges  that  can  be  marked  on  the  plank  as  shown 
by  the  short  broken  lines;  again  on  the  lower  side  of  the  plank  (on  the  lines  1,  2  and  T  B)  move  the 
face-mould  as  much  this  way  from  1  and  T,  marking  joint  B  and  point  B,  and  the  edges  of  the  face- 
mould  that  remain  on  the  plank  as  before.  Cut  away  the  wood  and  make  the  plumb-joint  from  B  as 
indicated  on  both  faces  of  the  plank  ;  then  mark  a  line  on  the  plumb-joint  from  the  points  B,  B  at  the 
upper  and  lower  faces  of  plank  ;  and  this  will  be  the  plumb  line  from  which  to  square  the  wreath  at  this 
joint.     No  straight  wood  is  required  at  the  upper  end  except  that  added  to  plumb  the  joint. 

Fig.  5.  Construction  of  a  Paper  Representation  of  a  Solid  which  Presents  a  Practical  Dem- 
onstration in  Solid  Geometry  of  the  Laws  Controlling  the  Tangents,  the  Trace  for  Face- 
mould,  and  the  Position  of  the  Plane  of  the  Plank  where  the  Plan,  a  Semicircle,  is  to  be  Cov- 
ered in  one  Wreath-piece. —  Let  J  C  Z  equal  J  C  Z  of  Fig.  i  ;  at  right  angles  to  J  C  draw  C  D  ;  at  right 
angles  to  J  Z  draw  Z  7  ;  at  right  angles  to  Z  C  draw  C  X  and  Z  W  ;  make  C  D,  C  X,  Z  7  and  Z  W  each 
equal  C  D  of  Fig.  i  ;  connect  D  J,  7  J  and  W  X.  These  form  the  three  vertical  sides  of  the  solid,  and 
show  in  position  the  inclined  and  level  tangents,  also  the  inclination  of  the  plane  of  the  plank  over  the  diam- 
eter of  the  cylinder;  showing  also  that  the  point  U  of  the  centre  line  of  rail  must  invariably  fall  below  the 
position  of  the  level  tangent  X  B.  To  Find  the  Am^^les  Enclosing  the  Cutting  Plane,  also  the  Trace  of  the  Given  Semi- 
circle :—FdYa\\Q\  to  Z  C  draw  8,  1 4,  T  P,  4  Q,  1 6,  N  and  2.  1  5  ;  parallel  to  C  D  draw  1 4,  1  3,  P  E,  Q  F,  N  G  and 
15,  1  ;  at  right  angles  to  Z  C  draw  J  A,  1  6,  1  8  and  T  V  indefinitely.  On  X  as  centre  witli  J  D  as  radius  describe 
an  arc  at  A  ;  on  W  as  centre  with  J  7  as  radius  intersect  the  arc  at  A;  connect  X  A  W.  Make  X  1  1 ,  1 0,  9  Y  S  A 
equal  D  13  E  F  G  IJ  ;  parallel  to  X  W  draw  S  R,  Y  0,  9  M,  1  0,  V  and  1 1  K  ;  make  S  R  equal  15,  2,  and  Y  0  equal 
N  3,  and  9  M  equal  Q  4,  and  10,  L  equal  P  5,  and  11  H  K  equal  14,  6,  8  ;  through  VKBHLMORA  find  the 
trace  of  the  semicircle  J  5  T.  With  a  sharp  pointed  instrument  scratch  the  lines  of  the  baseZ  J,  J  C  and 
CZ;  also  the  level  line  W  X.  Cut  out  the  remainder  of  the  figure,  touch  the  adjoining  edges  with  a 
little  glue    or  thick    mucilage,  and  bring  them   together  so    as  to  leave  the  shading  and  lines  outside. 

Fig  6.  To  Develop  or  Unfold  the  Semicircle  J  5  T  of  Fig.  5,  and  its  Vertical  Intersection 
with  the  Cutting  Plane  of  that  Figure  at  the  Points  V  K  B  H  L  M  0  R  A:— Draw  the  line  J  T  an.i 
mark  on  it  tiie  divisions  as  shown  by  the  corresponding  letters  and  figures  on  the  semicircle  at  Fig.  5. 
Raise  perpendiculars  at  each  of  these  divisions  equal  in  height  to  those  indicated  by  similar  letters  at 
Fig.  5;  and  through  these  trace  the  points  of  interse'ction  as  shown.  This  curve,  J1GFE13D13E  is 
the  unfolded  centre  line  of  wreath  the  plan  of  which  is  given  at  Fig.  i.  As  the  point  E  is  the  centre  of 
the  rail  joint  at  the  upper  end,  it  is  evident  that  considerable  wood  will  have  to  be  cut  away  at  D  in 
shaping  the  wreath  to  a  level  ;  but  the  slab  that  is  removed  from  the  top  may  be  glued  to  the  bottom. 

In  this  treatment  of  a  wreath  in  one  piece,  the  longer  the  tangent  J  C,  Fig.  i,  the  better ;  and  I  might  also 
add  the  smaller  the  cylinder  the  better.     See  Pla  te  No.  79. 


PLATE  35. 


Another  and  More  Simple  Method  of  Drawing  a  Face-mould  and  Working  the  Whole 
Wreath  in  One  Piece  over  a  Seven-inch  Cylinder. — Size  of  Hand-rail  Four  Inches  Wide 
HY  Two  Inches  and  a  Quarter  Thick  ;  the  Wreath  to  he  Got  out  of  Plank  Four  Inches 
Thick. 

Fig.  I.  Plan  of  the  top  porlion  of  a  straight  flight  of  stairs  with  a  7"  cylinder,  the  landing- 
riser  placed  at  the  cliord-line  or  commencement  of  the  cylinder. 

Fig.  2.  After  drawing  the  plan  as  given  and  described,  set  up  this  tread  and  rise  in  a  con- 
venient position  to  the  plan  as  shown.  Rest  the  bottom  of  the  rail  at  X  X,  the  centres  of  the  short 
balusters.  The  bottom  of  the  level  rail  B  is  set  up  from  the  floor  4"  as  usual;  draw  DB  parallel 
to  the  floor-line;  let  BH  equal  the  thickness  of  the  rail;  make  HK  parallel  to  B  D;  at  E  draw  EGN 
at  right  angles  to  XX;  thiongh   G   parallel  to  the  rise-line  draw  J  K. 

Fig.  I.  To  Draw  the  Face-mould,  also  to  Find  the  Squaring  of  the  Wreath  at  its 
Joints: — At  right  angles  to  AB  draw  B  2,  0  W,  S  Y,  E  3,  P  T,  A  Z,  etc.,  indefinitely;  parallel  to  the 
floor-line,  FiG.  2,  from  0  and  C  draw  lines  lo  L  and  C  of  Fig.  i,  and  again,  parallel  to  LO  or  C  C 
draw  NJ  F,  G  Z,  B  D  6,  and  H  K  Y.  At  Fig.  i  make  LZ  and  LD  each  equal  half  the  thickness— 2"— 
of  plank;  parallel  to  LC  draw  Z2  and  D  8,  showing  the  edge  of  the  plank  as  canted  in  its  required 
position;  touching  the  angles  U,  6,  X,  and  F,  draw  lines  at  right  angles  to  Z  2.  The  square  shaded 
sections  are  the  joints  of  the  wreath  as  cut  square  through  the  plank,  showing  also  on  these  joints 
the  lines  that  square  the  wreath.  For  straight  wood  required  to  make  the  butt  joint  of  the  lower 
end:  LetTK  equal  E  0  of  Fig.  2;  through  K  parallel  to  LC  draw  5  J.  At  right  angles  to  Z  2  draw 
2  J,  W  M,  TV,  Z  5,  etc.,  measuring  all  the  points  for  tracing  the  edges  of  the  face-mould  from  the 
line  A  B,  and  setting  them  off  from  the  line  J  5;  as  I  M  equals  OQ;  KV  equals  P  R,  etc.  Mark  the 
face-mould  on  the  plank,  and  saw  out  square;  plane  the  joints  square  from  the  face  of  the  plank; 
this  done,  lay  out  each  joint  for  squaring  the  wreath  with  the  bevel  W  as  shown.  Before  making 
the  butt  Joint  at  L,  cut  off  at  right  angles  to  the  joints  the  slabs  UY  and  D  F;  then  resting  D  F  on 
the  saw-table,  let  the  band-saw  cut  each  side  of  the  wreath  plumb  as  far  as  the  centre;  reverse  the 
ends;  this  time  resting  the  level  surface  U  Y  on  the  saw-table  and  finish  sawing  plumb  the  sides 
of  the  remaining  half  of  the  wreath. 

To  Make  the  Butt  Joint  at  L,  as  at  G  E  of  Fig.  2:— Gauge  from  the  joint  on  the  level 
surface  of  F  D  the  distance  J  N  of  Fig.  2;  then  cut  away  the  superfluous  wood  and  make  the  joint 
required  from  the  gauged  line  to  the  line  X  Z.  The  butt  joint  made  and  tested*  and  the  sides 
sawed  plumb,  finish  the  squaring  by  band-saw,  cutting  away  for  the  top  and  bottom  of  the  wreath, 
rolling  it  on  the  saw-table  on  its  convex  side.  To  increase  the  thickness  of  plank  at  the  point 
most  required,  the  slab  sawed  off  along  the  line  UY  at  the  top  may  be  glued  on  at  Z  T  of  the 
lower  end  L. 

By  this  method  of  managing  a  wreath  in  one  piece,  where  the  plan  is  arranged  with  the  riser 
set  into  the  cylinder  as  at  Plate  34,  the  only  difference  will  be  that  the  plank  will  have  more  cant 
or  inclination;  but  as  a  wreath  it  will  work  and  answer  equally  as  well. 

Side  Moulds. — For  a  better  understanding  of  this  case  of  hand-railing  it  will  be  useful  to 
unfold  the  side  moulds, f  as  follows: — Let  F  N  of  Fig.  i,  equal  S  C  of  Fig.  2;  connect  N  R;  divide  the 
centre  line  P6  0  into,  say  six  equal  parts,  and  draw  radial  lines  through  each  of  these  six  divisions 
as  shown;  again,  through  the  same  divisions  draw  lines  parallel  to  0  N  as  3,  3;  X,  5;  6,  6,  etc. 

To  Unfold  the  Convex  Side  Mould: — Fig.  3.  On  a  line  FA  mark  the  six  divisions  taken 
around  BA  of  Fig.  1;  take  the  heights  F  N,  3,  3,  etc.,  of  Fig.  i  and  place  them  at  the  corresponding 
letters  and  figures  of  Fig.  3;  through  the  points  thus  found  trace  the  dotted  lines  which  unfold  the 
natural  form  of  the  side  mould;  but  this  form  and  joint  requires  to  be  forced, — changed  some  to 
make  it  properly  connect  with  the  hand-rail  of  the  flight.  Let  A  B  equal  the  rise  and  B  C  the  tread; 
connect  C  A;  through  E  make  the  joint  line  D  E  0  at  right  angles  to  A  C;  from  D  draw  D  X  parallel  to 
A  B.  When  this  convex  side  mould  is  applied  to  the  wreath,  D  X  agrees  with  the  joint  Z  F,  and  N 
with  the  joint  U,  both  of  Fig.  i. 

Concave  Side  Mould: — Fig.  6.  The  line  A  F  shows  the  six  divisions  taken  around  E  S,  Fig.  i, 
and  the  heights  are  the  same  as  at  Fig.  3;  the  joints  are  also  the  same. 

WAINSCOT. 

Fig.  4.  Vertical  Cross-section  of  Hall  Wainscot,  of  which  an  Elevation  of  a  Portion 
is  made  at  Fig.  5,  as  it  Appears  along  the  Wall,  in  Connection  with  the  Level  Wainscot 
at  the  Starting  of  a  Staircase. — In  the  best  panel-work  of  hard  wood,  the  frame  is  put 
together,  the  mouldings  glued  in  place,  and  the  whole  finished  and  varnished.  Afterwards  the 
varnished  panels  are  set  in  from  the  back  and  fastened  at  XX  with  three-cornered  pieces  of 
stout  sheet-metal  driven  in  the  frame;  or  in  place  of  the  metal,  strips  of  wood  aie  nailed  into 
the  frame  and  against  the  panels  at  X  X.  As  to  the  height  wainscot  should  be  up  the  flight 
as  compared  with  the  height  of  the  connecting  level  wainscot,  there  is  no  fixed  rule;  but  as  a 
basis,  let  A  B  of  Fig.  5,  the  level  wainscot,  be  three  feet  in  height  as  at  Fig.  4.  Make  a  vertical 
line  along  any  riser  of  the  flight  C  D  equal  three  feet;  then  if,  after  measuring  on  the  line  H  J  — 
which  is  drawn  at  right  angles  to  the  inclined  string — the  width  of  bottom  rail  E,  middle  rail  F, 
and  top  rail  G,  as  given  at  the  section.  Fig.  4,  the  space  left  for  panels  is  unpleasantly  narrow, 
changes  may  be  made  and  the  height  given  at  C  D  altered  to  suit. 


*  The  butt  joint  at  L  may  be  tested  with  the  pitch  board,  when  the  level  surface  U  Y  of  the  wreath  is  made  to  rest 
on  the  saw-table.       f  See  Side  Moulds,  Platk  No.  76. 


Plate  No.  35 


Fig. 4. 

Scale  V4  In.  =  1  Ft. 


PLATE  36. 


Fig.  I,  Plan  of  a  Winding  Stairs  Turning  One  Quarter  at  or  about  the  Middle  of  the 
Flight. — Tliis  plan  is  given  at  Platk  No.  5,  Fig.  ii.  After  making  the  plan,  describe  the 
centre  line  of  the  rail  Q  H,  the  plan  tangents  Q  C  and  C  H.  Place  the  centres  of  balusters  as 
required.  Before  the  plan  can  be  completely  prepared  to  draw  a  parallel  pattern  or  a  face- 
mould,   the  elevation   must  be  drawn. 

Fig.  2.  Elevation  of  Plan  as  given  at  Fig.  I.— Set  up  the  treads  and  rises  as  figured  and 
given  at  the  plan  according  to  the  scale.  The  treads  in  the  cylinder  must  be  measured  on  the 
centre  line  of  the  rail,  and  each  tread  taken  in  two  parts  for  the  purpose  of  getting  more  accu- 
rately the  stretch-out  of  the  centre  line.  Place  the  centres  of  balusters  on  each  tread  as  shown 
on  the  plan,  and  except  at  the  centres  of  the  short  balusters  0,  draw  lines  parallel  to  the  rise- 
lines,  and  indefinitely.  At  the  upper  portion  of  the  elevation  through  the  centres  of  the  short 
balusters  0  0  draw  the  bottom  line  of  the  rail,  and  place  the  centre  line  N  C  in  position  par- 
allel to  0  0,  but  indefinite  in  length.  Anywhere  along  the  upper  chord-line  set  of?  the  length 
of  plan  tangent  H  C  of  Fig.  i,  and  draw  the  line  C  M  parallel  to  the  chord-line;  and  where  the 
centre  line  N  A  intersects  at  C,  draw  the  line  C  D  at  right  angles  to  the  chord-line.  Anywhere 
along  the  lower  chord-line  P  B  set  off  the  length  of  the  plan  tangent  Q  C  of  Fig.  i  and  draw 
the  line  F  E.  E  is  a  fixed  point  from  which  the  line  E  L  may  be  drawn  to  suit  its  position  over 
the  winders  and  the  requirements  of  the  ramp.  Wherever  the  line  E  L  intersects  the  chord- 
line  P  B  as  at  B,  draw  the  line  B  F  at  right  angles  to  the  chord-line.  Make  A  J  equal  four 
inches  for  straight  wood  on  the  upper  end  of  the  wreath-piece;  and  make  B  G  also  four  inches 
for  straight  wood  on  the  lower  end.  The  ramp  is  curved  as  shown.  Again  at  Fig.  1  make 
C  E  at  right  angles  to  Q  C  and  equal  to  F  E  of  Fig.  2;  connect  E  Q;  make  H  D  at  right 
angles  to  C  H   and  equal  to   D  A  of  Fig.  2;  connect  D  C. 

To  Find  the  Directing  Level  Line: — Make  CG  equal  H  D;  make  G  F  parallel  to  C  Q, 
and  F  0  parallel  to  C  E;  connect  0  N,  which  is  the  level  line  sought.  Parallel  to  N  0  draw 
IZ,  UO,  2T,  CM  and  3  W;  parallel  to  C  E  draw  TX  and  Z  Y;  parallel  to  H  D  draw  W  K; 
from  H  at  right  angles  to  NO  draw  H  A  indefinitely;  from  Q  at  right  angles  to  N  0  draw 
Q  B  indefinitely;  on  C  as  centre  with  CD  as  radius  describe  the  arc  D  A;  again  on  C  as 
centre  with   E  Q  as  radius  describe  an   arc  at   B;  connect   B  A. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  its  Joint  over  H: — 
Make  H  L  equal   H  J;  connect  L  M;  then  the  bevel  at  L  will  give  the  angle  required. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  its  Joint  over  Q:— Pro- 
long H  N  to  R  indefinitely;  make  N  R  equal  0  P;  connect  R  Q;  then  the  bevel  at  R  will  give  the 
angle  sought. 

To  Develop  the  Centre  Line  of  the  Wreath-piece  in  Position  over  the  Elevation, 
Fig.  2. —  Make  Z  Y,  T  X  and  W  K  equal  the  heights  at  the  corresponding  letters  of  Fig.  i;  then 
through  B  Y  X  K  A  trace  the  centre  line  of  the  wreath.  Set  off  half  the  thickness  of  rail  each 
side  of  the  centre  as  shown  by  the  dotted  lines. 

To  Find  the  Length  of  Balusters: — Take  for  exam.ple  the  one  marked  3;  3  V  measures 
5:^",  which,  added  to  2'. 2",  the  length  of  short  baluster  at  0,  equals  2'.-jY  between  the  top  of 
the  step  3  and  the  bottom  of  the  rail  V  at  the  centre  of  the  baluster. 

Fig.  3.  Parallel  Pattern  for  the  Wreath-piece  over  the  Plan,  Fig.  i.— Make  the  line 
B  D  equal  B  A  of  Fig.  i;  make  D  S  equal  A  S  of  Fig.  i;  on  D  as  centre  with  C  D  of  Fig.  i  as 
radius  describe  an  arc  at  C;  and  on  B  as  centre  with  Q  E  of  Fig.  i  as  radius  intersect  the  arc 
at  C;  also  on  S  as  centre  with  S  C  of  Fig.  i  as  radius  test  the  intersection  of  the  arcs  at  C;  con- 
nect D  C,  C  B  and  S  C;  prolong  C  B  to  G  and  CD  to  J ;  make  B  G  equal  B  G  of  Fig.  2;  make 
D  J  equal  A  J  of  FiG.  2;  make  B  Y  F  X  equal  Q  Y  F  X  of  Fig.  i;  make  C  K  equal  C  K  of  Fig.  i; 
parallel  to  the  level  line  C  S  draw  K  3,  X  2,  F  U  and  Y  I;  make  Y  I,  F  U,  X  2  and  C  V  equal  Z  I. 
0  U,  T  2  and  C  V  of  Fig.  i;  make  K  3  equal  W  3  of  Fig.  i.  The  joints  J  and  G  are  made  at 
right  angles  to  the  tangents. 

At  a  trial — laying  out  the  squaring  of  the  wreath-piece  at  joint  J  with  the  bevel  L  of  Fig.  i, 
also  joint  G  with  bevel  R  of  Fig.  i — it  is  found  that  the  position  of  the  form  of  rail  at 
joint  J  takes  the  greatest  width  of  stuff,  equal  to  5",  therefore,  with  2^"  radius  describe  circles 
on  the  centres  GB1  U2V3DJ,  and  trace  lines  touching    the  circles  to  complete  the  pattern. 

Development  of  the  centre  line  of  wreath-piece  in  cases  of  this  kind  is  given  in  detail  at 
Plate  No,  20  by  the  quarter-circle  Q  V  of  Fig.  3.  Face-mould  and  parallel  pattern  of  this 
character  is  treated  in  detail  at  Plate  No.  12. 


PLATE 


Fig.  I.  Plan  of  a  Half-turn  Platform  Stairs  with  the  Opening  between  the  Strings— 
Usually  Built  to  Connect  in  the  Form  of  a  Semicircle — Composed  of  Two  Quarter-circles 
with  Straight  between. — Plans  of  platform  stairs  differently  treated  are  given  at  Plates  6 
and  7.  The  situation  of  the  risers  in  connection  with  the  chord  lines  is,  in  this  case,  determined 
by  trial  through  the  elevations  of  tread  and  rise  set  up  at  Figs.  2  and  3. 

Fig.  2. — Let  the  bottom  line  of  hand-rail  pass  through  X  X,  the  centres  of  short  balusters; 
the  thickness  of  rail  is  set  off  parallel  to  X  X  by  the  line  E  D;  the  line  C  B  is  the  centre  of 
a  four-inch  plank,  from  which  the  wreath-piece  is  to  be  worked  out.  A  J  is  four  inches,  which 
the  rail  is  to  rise  above  the  floor  more  than  the  height  to  be  raised  at  XX;  J  B  is  half  the 
thickness  of  rail,  and  the  height  of  B,  touching  the  centre  line  C  B,  determines  the  exact 
position  of  B  to  the  riser  H;  and  at  the  plan  Fig.  i  A  H  is  made  to  equal  A  H  of  Fig.  2. 
This  explanation  and  the  corresponding  letters  will  sen-e  Fig.  3.  At  Fig.  i,  to  prepare  for  drawing 
the  face-mould,  place  the  pitch-board  as  shown,  marking  the  line  of  hypothenuse  W  M;  prolong 
U  A  to   M;  parallel  to   U  M   draw  T  N  and  V  0. 

Fig.  4.  Face-mould  from  Plan  Fig.  i;  also  the  Squaring  of  Wreath-piece  at  Both 
Joints. — Draw  U  A  and  A  K  at  right  angles;  make  A  I  LU  equal  A!  LU  of  Fig.  i;  make  A  N  OW 
equal  M  N  0  W  of  F"ig.  i;  make  W  K  four  inches  for  straight  wood;  through  N,  0,  W,  K,  parallel 
to  U  A,  draw  Y  Z,  IV,  F  E  and  B  C;  make  W  F  and  W  E  each  equal  the  same  at  Fig  i;  make 
0  1  and  0  V  equal  PI  and  P  V  of  Fig.  i;  make  NY  equal  S  Y  of  Fig.  i;  parallel  to  W  K 
draw  F  B  and  E  C;  make  L  J  equal  L  T;  parallel  to  L  U  draw  J  R;  through  U  draw  R  Z 
parallel  to  AW;  through  F  1  Y  I  J  of  the  convex  and  E  V  T  of  the  concave  trace  the  edges  of 
the  face-mould.  The  squaring  of  the  centre  joint  U  by  the  use  of  the  pitch-board,  as  shown, 
is  a  sufficient  explanation.  At  joint  K  the  sides  of  the  rail  are  at  right  angles  to  the  plane 
of  the  plank;  D  H  S  S  is  the  over-wood  to  be  cut  away  at  the  bottom  of  the  lower  wreath- 
piece,  and  at  the  top  of  the  upper  one. 

Fig.  5.  Plan,  as  given  at  Plate  5,  Fig.  10,  of  a  Quarter-platform  Stairs  turning 
One  Quarter  with  a  Quarter-cylinder. — Draw  the  plan  of  rail,  also  the  centre  line,  and  upon 
tlie  latter  space  the  balusters,  as  required.  Make  the  tangents  to  the  centre  line  of  the  rail, 
B  F  and  B  0,  and  let  the  distance  from  the  angle  B,  both  ways  to  each  riser,  equal  half  a 
tr.;ad — 4^"  from  B  to  S  and  4^"  from  B  to  1 .  By  this  arrangement  there  is  between  the  two 
risers  one  tread,  which  brings  the  wreath-piece  on  a  common  inclination  with  the  flight,  and 
makes  the  best  possible  shape  of  it.  To  further  prepare  the  plan  for  drawing  the  face-mould, 
place  the  tread  of  the  pitch-board  on  the  tangent  B  0  and  mark  the  line  B  Q;  prolong  the 
tangent  0  B  to  C;  prolong  A  0  to  Q.  A  line  connecting  A  B  will  be  the  level  line.  Parallel 
to  A  B  draw  R  L  and  H  G;  parallel  to  0  Q  draw  X  N  and  1  M;  parallel  to  B  C  draw  G  E; 
from   F  through  0  draw   F  P;    on   B  as  centre  with   B  Q  for  radius  describe  the  arc  Q  P. 

To  Find  the  Angle  with  which  to  Square  Both  Joints  of  the  Wreath-piece: — On  B 
as  centre  describe  an  arc  touciiing  the  line  C  F  and  D;  connect  D  F:  then  the  bevel  at  D 
will  give  the  angle  required. 

Fig.  6.  Elevation  of  Tread  and  Rise,  including  the  Platform,  as  given  at  Plan  and 
as  Figured. — The  platform  is  measured  at  the  plan  on  the  centre  line  in  two  parts  from  riser 
to  riser.  The  heights  and  inclinations,  the  rail  above  and  below,  the  joints  Y  and  Y,  are  all 
shown  in  position;  also  the  development  of  the  centre  line  of  wreath.  The  letters  correspond 
wiih  the  plan  Fig.  5  and  with  those  at  the  joints  of  the  face-mould.  The  length  of  baluster 
is  here  determined  as  before  explained. 

Fig.  7.  Face-mould  over  Plan  Fig.  5  ;  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  the  Joints. — Draw  the  line  P  F  indefinitely;  make  K  P  and  K  F  each  equal  K  P  of 
Fig.  5;  from  K  at  right  angles  to  F  P  draw  K  C;  make  K  C  equal  K  B  of  Fig.  5;  connect 
C  P  and  C  F;  prolong  C  F  and  C  P  to  Y;  make  P  Y  and  F  Y  equal  the  same  ai  Fig.  6;  make 
the  joints  Y,  Y  at  right  angles  to  the  tangents;  make  F  M  and  P  M  each  equal  Q  M  of  Fig.  5; 
through  M  and  M  parallel  to  C  K  draw  L  R,  L  R;  make  M  L  and  M  R  equal  1  L  and  1  R  of 
Fig.  5;  make  CZ  equal  B  Z  of  Fig.  5;  through  F  and  through  P  draw  R  V;  make  FV  equal 
F  R,  and  P  V  equal  P  R;  parallel  to  C  Y  draw  V  X  and  R  X  at  each  end;  through  V  L  Z  L  V  of 
the  convex  and  R  K  R  of  the  concave  trace  the  edges  of  the  face-mould.  The  angle  for 
squaring  the  wreath-piece  at  both  joints  is  given  by  the  bevel  D,  Fig.  5.  The  face-mould, 
Fig.  7,  is  treated  in  detail  at  Plate  No.  ii,  and  face  mould,  Fig.  4,  is  also  treated  in  detail  at 
Plate  No.  10. 


Plate  No. 37 


Plate  No.  38 


P  LATFO  R  M 


I 


PLATE  38. 


Fig.  I.  Plan  of  a  Platform  Stairs  with  the  Risers  at  Platform  set  in  the  Whole 
Depth  of  the  Cylinder. — Draw  the  centre  line  of  rail  and  space  the  balusters  as  required; 
also  draw  O  A  and  A  Q,  F  G  and  G  C  tangents  to  the  centre  line  of  rail  at  each  quarter- 
circle.  To  prepare  tlie  plan  for  measurements  that  will  develop  the  centre  line  of  wreath,  or 
to  draw  the  face-mould,  tlie  elevation  must  first  be  set  up. 

Fig.  2.    Elevation  of  Treads  and  Rises  as  given  at  the  Plan  and  as  Figured ;  also 
Development  of  the  Centre  Line  of  Wreath, — Let  tlie  bottom  line  of  rail  above  and  below 
the  platform  pass  through  X  X,  the  centres  of  short  balusters.    Place  the  chord-lines — of  which 
there  are  four — as  given  on    the  plan.    Draw  the  centre  line  of    rail   L  B  and  G  R  parallel  to 
X  X;  at  right  angles  to  the  chord-line  D  0  draw  0  A  equal  to  the  first  tangent  0  A  of  Fig.  i. 
Parallel  to  D  0  draw  A  B;  at  right  angles  to  A  B,  touching  B,  draw  Q  T;  make  QT  equal  the 
second  tangent   A  Q  of   FiG.   i;  make  C  G  at    right  angles  to    C  M  and    equal  to  the  tangent 
G  C  of  Fig.  i;  make  FE  equal  FG  of  Fig.  i,  and  draw  the  line  EZ  parallel  to  J  F;  from  Z  draw  the 
line  ZT;  where  the  inclined  line  ZT  intersects  the  chord-line  J  F  at  F,  draw  F  E  at  right  ang'es  to 
J  F.    Place  the  balusters  numbered  1,  2,  3,  4,  5,  6  as  on  the  plan,  and  draw  a  line  through  the  place 
of  each  baluster  parallel  to  the  rise-line  and  indefinitely. 

To  Prepare  the  Plan,  Fig.  i,  for  Finding  the  Lengths  of  Balusters :— Make  C  M  G 
equal  the  same  of  Fig.  2;  make  G  Z  F  equal  E  Z  F  of  Fig.  2;  make  Q  N  A  equal  Q  N  f  of 
Fig.  2;  make  ABO  equal  the  same  at  Fig.  2;  make  A  J  equal  QN;  make  JY  parallel  to 
A  0,  and  Y  8  parallel  to  B  A;  connect  8  R;  parallel  to  8  R  draw  2  V;  parallel  to  Q  N  draw 
V  W:  make  M  H  equal  G  Z;  draw  H  E  parallel  to  C  G,  and  ET  parallel  to  C  M;  connect  T  S; 
parallel  to  S  T  draw  5  K  and  4,  1;  parallel  to  C  M  draw  K  U;  parallel  to  G  Z  draw  I  P.  The 
heights  for  the  balusters  are  taken  as  numbered  and  lettered,  and  set  up  at  the  elevation,  Fig.  2, 
as  designated  by  the  same  numbers  and  letters;  then  tb'-'^ugh  these  letters— M  U  P  F  N  W  Y  0— 
trace  the  centre  line  of  the  wreath-pieces.  Set  off  each  side  of  the  centre  half  the  thickness  of 
rail  as  shown  by  the  dotted  lines;  next,  ge(  the  length  of  balusters:  take  for  example  baluster 
4— this  measures  from  the  platform  to  the  bottom  of  the  rail,  4!",  which,  added  to  2'.2" — the 
length  of  short  balusters  at  X  X — equals  2'.6f"  from  the  top  of  the  platform  to  the  bottom  of  the 
wreath  along  the  centre  line  of  baluster. 

Fig.  3.  Plan  of  the  Lower  Quarter  Wreath-piece.— The  heights  A  B  and  Q  N  are  taken 
from  those  lettered  the  same  at  Fig.  2.  Make  A  F  equal  Q  N;  draw  F  E  parallel  to  AO,  and  E  D 
parallel  to  B  A;  connect  C  D,  which  is  the  directing  level  line;  parallel  to  C  D  draw  V  W,  A  K,  R  S 
and  Q  Y;  at  right  angles  to  the  level  line  C  D  from  0  and  Q  draw  0  M;  also  Q  P,  each  indefi- 
nitely. On  A  as  centre  with  B  0  as  radius  describe  an  arc  at  M;  again  on  A  as  centre  with  A  N 
as  radius  describe  the  arc  N  P,  connect  P  M. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  0:— 
Prolong  Q  C  to  H;  make  C  H  equal  D  I;  connect  H  0;  then  the  bevel  at  H  will  give  the  angle 
required. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  Q  :— 

Prolong  A  Q  to  L;  make  Q  L  equal  Q  8;  connect  L  K;  then  the  bevel  at  L  contains  the  angle 
sought. 

Fig.  4.  Face-mould  taken  from  Plan  of  Quarter-circle  Fig.  3,  also  Showing  the 
Squaring  of  the  Wreath-piece  at  Both  Joints.— Make  0  N  J  equal  M  J  P  of  Fig.  3.    On  N 

as  centre  with  N  A  of  Fig.  3  as  radius  describe  an  arc  at  B;  on  0  as  centre  with  0  B  of  Fig.  3  as 
radius  describe  an  intersecting  arc  at  B;  on  J  as  centre  test  the  intersection  at  B  with  J  A  of  Fig.  3 
as  radius;*  connect  N  B,  0  B  and  J  B;  prolong  0  L  equal  to  0  L  of  Fig.  2;  and  prolong  B  N 
equal  to  N  S  of  Fig  2.  Make  N  T  equal  N  T  of  Fig.  3;  make  0  X  E  equal  0  X  E  of  Fig.  3; 
parallel  to  the  level  line  J  B  draw  N  Y,  T  R  ,  E  G  U  and  X  V;  make  N  Y  equal  Q  Y  of  Fig.  3; 
make  T  R  and  B  7  Z  equal  S  R  and  A  7  Z  of  Fig.  3;  make  E  G,  E  U  equal  D  G,  D  U  of  Fig.  3; 
make  X  V  equal  W  V  of  Fig,  3;  through  N  draw  R  F;  make  N  F  equal  N  R;  make  the  joints  S  and 
L  at  right  angles  to  the  tangents;  from  R  and  F  draw  lines  to  the  joint  parallel  to  N  S;  through  0 
draw  V  D;  make  0  D  equal  0  V;  through  F  Y  7  G  D  of  the  convex  and  R  Z  U  V  of  the  concave 
trace  the  curved  edges  of  the  face-mould.  The  slide-line  is  made  at  right  angles  to  the  level 
line  J  B.  The  angle  for  squaring  the  wreath-piece  at  joint  L  is  taken  by  the  bevel  at  H  of 
Fig  3,  and  that  for  the  centre  joint  S  is  taken  by  the  bevel  at  L  of  Fig.  3.  This  face-mould 
is  treated  in  detail  at  Plate  No.  12.  The  development  of  the  centre  line  of  wreath-piece  in 
a  case  of  this  kind  is  shown  in  detail  at  Plate  No.  20,  plan  of  quarter-circle  R  Q  V,  and 
Fig.  4,  Q  to  Y. 


*  Instead  of  using  the  length  of  the  level  line  J  A  of  Fig.  3,  as  at  J  B  of  Fig.  4,  as  a  test,  it  may  be  used  in  any 
case,  together  with  the  length  of  either  one  of  the  tangents,  to  establish  the  angle  of  tangents  as  at  B. 


PLATE 


39 


Fiff.  I.  Plan  of  Stairs  with  Two  Connecting  Platforms  Divided  by  a  Riser,  set  in 
^he  Centre  of  the  CyHnder  in  a  Direction  Parallel  to  the  Strings;  also  with  the  Risers 
Landing  on  and  Starting  from  the  Platforms,  Each  set  into  the  Cylinder  Three 
Inches.* — Set  off  the  centre  line  of  rail  and  space  the  balusters  as  required;  also  draw  the 
ti'.norents  0  C,  C  2  and  A  F,  F  H  to  the  centre  line  of  the  rail  at  each  quarter-circle;  then  to 
find  the  angles  of  inclination  for  the  tangents  it  is  necessary  first  to  set  up  the  elevati(m. 

Fig.  2.  Elevation  of  Treads  and  Risers  as  given  at  the  Plan  and  as  Figured; 
also  the  Development  of  the  Centre  Line  of  Wreath-pieces. — Let  the  bottom  lines  of  rail 
above  and  below  the  platforms  pass  through  X  X,  the  centres  of  short  balusters.  Place  the 
chord-lines,  of  which  there  are  four,  as  given  on  the  plan;  draw  the  centre  line  of  rail  R  E 
and  T  G  parallel  to  X  X;  at  right  angles  to  the  first  chord-line  W  D  draw  D  C  equal  to  the 
first  tangent  D  C  of  Fig.  i;  parallel  to  W  D  draw  C  E;  touching  E,  at  right  angles  to  2  B, 
draw  M  Q  equal  to  the  second  tangent  2  C  of  Fig.  i:  then  Q  becomes  a  fixed  point.  At 
the  uppermost  chord  draw  H  G  at  the  intersection  of  the  centre  line  G  T,  equal  to  the  fourth 
tangent  H  F  of  Fig.  i;  from  the  third  chord-line,  Y  A,  make  AF  equal  F  A  of  Fig.  i;  make 
F  N  parallel  to  Y  A:  then  N  becomes  the  next  higher  fixed  point.  Connect  the  two  fixed 
points  N  and  Q;  where  the  inclined  line  N  Q  intersects  the  chord-line  at  A  draw  A  F  at  right 
angles  to  Y  A;  divide  the  straight  between  the  two  quarters  B  A  in  two  equal  parts  at  S: 
then  S  will  be  at  the  centre  joint  of  wreath-pieces.  Place  the  balusters  numbered  1,  2,  3,  4,  5 
as  on  the  plan,  and  draw  a  line  through  the  place  of  each  baluster  parallel  to  the  rise-lines 
indefinitely. 

To  Prepare  the  Plan  Fig.  i  for  Finding  the  Length  of  Balusters : — As  the  angles 
of  inclination  in  this  case  happen  to  be  all  alike,  the  angle  DEC  of  Fig.  2  may  be  set  in 
place  over  each  tangent,  as  D  E  C,  C  B  2,  A  N  F  and  H  J  F;  connect  Q  F,  also  G  C.  From  the 
centre  of  baluster  5  draw  5  M  parallel  to  H  J ;  from  the  centre  of  baluster  4  draw  4  K  parallel 
to  Q  F;  parallel  to  F  N  draw  K  L;  parallel  to  G  C  from  the  centre  of  baluster  1  draw  1,  0; 
parallel  to  C  E  draw  0  P;  at  Fig.  2  make  0  P  equal  0  P  of  Fig.  i;  make  K  L  and  U  V  of 
Fig.  2  equal  K  L  and  5  M  of  Fig.  i;  then  through  D  P  B,  A  LVJ  of  Fig.  2  trace  the  centre 
line  of  wreath-pieces;  set  off  each  side  of  the  centre  line  half  the  thickness  of  rail  as  shown 
by  the-  dotted  lines. 

To  Find  the  Lengths  of  Balusters: — Take  for  example  No.  2  baluster,  where  2Z  equals 
4^",  which  added  to  2'.2",  the  length  of  short  balusters  at  X  X,  makes  the  length  of  that 
baluster  on  the  line  of  its  centre  from  the  top  of  step  to  the  bottom  of  rail  2'. 63". 

Fig.  3.  Plan  of  the  Lower  Quarter-circle  with  Tangents  and  Angles  of  Inclination 
Lettered  Alike  and  as  taken  from  Fig.  l. — From  K  draw  K  M  parallel  to  G  C;  from  L  draw 
LH  parallel  to  2  B;  through  D  and  2  draw  DA  indefinitely;  on  C  as  centre  with  C  B  as  radius 
describe  the  arc  B  A. 

To  Find  the   Angle  with  which   to   Square  the   Wreath-piece  at  Both  Joints  :— 

Prolong  C  2  to  F  indefinitely;  on  2  as  centre  describe  an  arc  touching  the  inclined  line  C  B  and 
at  F;  connect  FG:  then  the  bevel  at  F  contains  the  angle  required. 

Fig.  4.  Face-mould  from  Plan  Fig.  3  ;  also  Squaring  of  the  Wreath-piece  at  Both 
Joints. — Draw  the  line  A  A  indefinitely;  let  J  A  and  J  A  each  equal  J  A  of  Fig.  3;  make  J  C  at 
right  angles  to  J  A  and  equal  to  J  C  of  Fig.  3;  connect  OA  and  OA;  prolong  C  A  to  S  and  CA 
to  R,  each  indefinitely;  make  CH  and  CH  each  equal  C  H  of  Fig.  3;  make  AR  equal  D  R  or 
J  T  of  Fig.  2,  for  straight  wood;  make  AS  equal  B  S  or  A  S  of  Fig.  2,  this  being  one  half  the 
straight  between  the  quarter-circles  of  which  this  cylinder  is  composed;  through  H  and  H 
parallel  to  the  level  line  JO  draw  M  K,  M  K;  make  H  K  and  H  M  equal  LK  and  L  M  of  Fig  3; 
make  J  !  and  J  N  each  equal  J  I  and  J  N  of  Fig.  3;  through  A  at  both  ends  draw  K  W;  make 
A  W,  A  W  each  equal  A  K;  the  joints  S  and  R  are  made  at  right  angles  to  the  tangents;  from 
K  and  W  draw  lines  parallel  to  the  tangents,  touching  the  joints;  through  W  M  C  M  W  of  the 
c  onvex  and  K  N  K  of  the  concave  trace  the  curved  edges  of  the  face-mould  The  angle  with 
which  to  square  the  wreath-piece  at  the  centre  joint  S  and  joint  R  is  contained  in  the  bevel 
at  F  of  Fig.  3.  The  development  of  the  centre  line  is  explained  in  detail  at  Plate  No.  20, 
Fig.  I,  quarter-circle  A  C,  and  A  Y  E  of  Fig.  2.  This  case  of  face-mould,  Fig.  4,  is  given  in 
detail  at  Plate  No.  11.     This  plan  of  stairs  is  given  at  Plate  No.  7,  Fig.  2. 


*  A  riser  is  calculated  as  set  in  a  cylinder  three  inches  more  or  less  from  the  chord-line  to  the  face  of  the  regular 
tread,  and  not  from  the  chord-line  to  any  point  of  the  cylinder  to  which  the  riser  may  be  curved. 


Plate  No.  40 


Scale  1^2  U  =  1  Ft. 


PLATE  40. 


Fig.  I.  Plan  of  Half-turn  Platform  Stairs  with  15"  Cylinder,  the  Risers  Landing  on 
and  Starting  from  the  Platform  set  in  the  Cylinder  7^",  its  Whole  Depth. — This  plan  is 
given  at  Plate  No.  7,  Fig.  i.  Describe  the  centre  line  of  rail,  and  draw  the  tangents  A  E. 
E  X.  Space  the  balusters  as  required.  Before  proceeding  further  in  the  preparation  of  this  plan 
for  drawing  the  required  face-mould  it  is  necessary  to  set  up  the  elevation. 

Fig.  2.  Elevation  of  Tread  and  Rise  as  given  at  the  Plan  and  as  Figured. — Place 
the  chord-lines,  of  which  there  are  two,  in  position  as  shown  on  the  plan  Fio.  i.  Let  the 
bottom  line  of  rail  pass  through  the  centres  of  short  balusters  at  0  0,  and  draw  the  centre 
line  of  rail  D  C  and  G  K  each  parallel  to  0  0.  From  the  first  chord-line  A  make  A  E  equal 
A  E  of  Fig.  i;  parallel  to  a  rise-line  draw  EC;  at  A  draw  A  E  at  right  angles  to  the  chord- 
line;  make  J  G  equal  A  E;  draw  G  X  parallel  to  the  rise-line;  from  G  draw  G  J  at  right 
angles  to  G  X;  from  C  draw  C  X  parallel  to  the  line  of  platform;  divide  G  X  at  F  in  two  equal 
parts. 

To  Prepare  the  Plan  Fig.  i  as  Required  to  Measure  for  Drawing  a  Face-mould. — 

Prolong  X  E  to  N  and  to  C;  make  E  C  equal  E  C  of  Fig.  2;  connect  C  A;  prolong  J  X  to  F 
and  to  I;  make  X  F  equal  X  F  of  Fig.  2;  connect  F  E;  make  E  D  equal  X  F;  draw  D  B  parallel 
to  E  A;  make    B  K  parallel  to  E  C;  connect  K  J,  which  is  the    level  line;  parallel  to  J  K  draw 

Y  2;  parallel  to   J  A  draw  2  W;  parallel    to  J  K  draw  E  M,  U  V  and  X  S;  parallel    to  J  X  draw 

V  R;  at  right  angles  to  J  X  draw  A  H  and  X  Z,  both  indefinitely;  on  E  as  centre  with  E  F  as 
radius  describe  the  arc  F  Z;  again,  on  E  as  centre  with  C  A  as  radius  describe  an  arc  at  H; 
connect  H  Z. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  X: — 

Make  X  N   equal  XG;  connect  N  M;  then  the  bevel  at   N   will  contain  the  angle  required. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  A: — 
Make  J  I   equal   K  Q;  connect  I  A:  then  the  bevel  at  1   will  contain  the  angle  sought. 

Fig.  3.  Face-mould  from  Plan  Fig.  i,  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints. — Draw  the  line  Z  H;  make  Z  0  and  0  H  equal  the  same  at  Fig.  i; 
on  Z  as  centre  with  F  E  of  Fig.  i  as  radius  describe  an  arc  at  C;  on  H  as  centre  with  A  C 
of  Fig.  I  as  radius  intersect  the  arc  at  C;  on  0  as  centre  with  0  E  of  Fig.  i  as  radius  test 
the  intersection  of  the  arcs  at  C;  connect  Z  C,  H  C  and  0  C;  prolong  C  H  to  D;  make  H  D 
equal  D  A  or  H  K  of  Fig.  2;  make  the  joints  D  and  Z  at  right  angles  to  the  tangents;  make 
H  W  B  equal  A  W  B  of  Fig.  i;  make  C  R  equal  E  R  of  Fig.  i;  parallel  to  0  C  draw  Z  S,  R  U, 
and  through  B,  L  4  and  W  Y;  make  WY  equal  2  Y  of  Fig.  i;  make  B  L  and  B4  equal  K  L 
and  K  4  of  Fig.  i;  make  C  P  equal  E  P  of  Fig.  i;  make  R  U  equal  V  U  of  Fig.  i;  make 
ZS  equal  X  S  of  Fig.  i.  Through  H  draw  Y  T;  make  HT  equal  YH;  through  Z  draw  UJ; 
make  ZJ  equal  Z  U;  draw  lines  from  T  and  Y  parallel  to  the  tangent  touching  joint  D. 
Through  T  L  P  S  J  of  the  convex  and  Y  4  0  U  of  the  concave  trace  the  curved  edges  of  the" 
face-mould.  The  slide-line  is  at  right  angles  to  the  level  line  0  C.  The  angle  for  squaring 
the  joint  Z  is  taken  by  the  bevel  N  of  Fig.  i,  and  for  squaring  joint  D  by  the  bevel  I  of 
Fig.  I. 

Fig.  4.  Elevation  of  Step  and  Rises  at  the  Starting,  Set  up  for  the  Purpose  of  Find- 
ing what  Position  the  Bottom  Riser  should  take  next  to  the  Chord-line  of  the  Cylinder 
when  the  \A/'reath-piece  is  Treated  in  the  Simplest  Manner,  with  the  Over-wood  all  to  be 
Removed  from  the  Top. — Let  the  bottom  line  of  rail  pass  through  the  centres  of  short  bal- 
usters X  X.  Set  off  the  thickness  of  rail  X  E,  and  draw  E  H  parallel  to  X  X;  let  X  D  be  the 
thickness  of  plank,  out  of  which  the  wreath-piece  is  to  be  worked.  Make  the  line  A  F  pass 
through  the  centre  of  X  D  and  parallel  to  XX;  make  B  C  four  inches  and  C  A  half  the  thickness 
of  rail.  The  intersection  of  the  line  A  J  at  its  given  height  with  the  centre  line  F  A  at  A  fixes  A 
as  the  centre  of  the  rail  at  the  centre  of  the  plank,  and  the  distance  from  A  to  the  chord-line  J 
must  be  8|-",  equal  to  J  A  of  Fig.  i,  or  D  K  of  Fig.  5. 

Fig.  5.  Plan  of  Cylinder  Connecting  Step  and  Hand-rail  at  the  Bottom  of  this  Flight 
of  Stairs.— Let  the  chord  line  H  of  the  cylinder  be  set  at  the  same  distance  from  the  bottom 
riser  as  shown  at  the  elevation  Fig.  4. 

To  Prepare  the  Plan  for  Drawing  a  Face-mould: — Draw  the  tangents  H  J  and  J  K; 
from  the  centre  D  describe  the  plan  of  rail.  Set  the  pitch-board  with  the  tread  on  the  line 
H  J,  and  mark  the  pitch-line  J  Q;  prolong  D  H  to  Q;  parallel  to  K  J   draw  L  N,  V  0  and   U  P 

Fig.  6.     Face-mould  from  Plan  Fig.   5,  also  Squaring  the   Wreath-piece  at  Both 
Joints. — Draw  Y  J,  J  K  at  right    angles;  make  J  N  0  PQ  equal  the  same  at  Fig.  5;  make  J  K 
equal  J  K  of   Fig.  5.    Through    K    parallel  to    J  Y    draw  X  L;   through    N  0  P  Q  Y    draw  lines 
parallel  to  J  K  indefinitely;  make  K  X  equal  K  L;  make  N  M,  0  W,  P  S  and  Q  R  equal  Y  M,  XW,  Z  S  and 
HR  of  Fig.  5.    Make  0  V,'  P  U  and  QT  equal  X  V,  Z  U  and  H  T  of  Fig.  5;  draw  lines  from  T  and  R 
to  the  joint  Y  parallel  to  QY;  through  X  I  M  W  S  R  of  the  convex  and  LVUT  of  the  concave  trace  the 
curved  edges  of  the  face-mould.    The  wreath-piece  at  the  joint  K  is  squared   by  the  use  of  the 
pitch-board  as  shown.    The  joint  at  Y   is  square,  and  E  D  is  the   over-wood  as  shown  at  E  D 
of  Fig.  4.    Face-mould  Fig.  6  is  treated  in  detail  at.  Plate  No.  10.    Face-mould  Fig.  3  is  explained 
in  detail  at  Plate  No.   12.     llie  dcveloptncnt  of  the  centre  line  at  FiG.  2  of  ivreath-pieces  is  given 
in  detail  at  Plate  No.  20,  Fig.  3,  quarter-circle  Q  V,  and  Fig.  4,  Q  Y, 


PLATE  41. 


Pig.  I,  Plan  of  Stairs  with  Two  Quarter  Platforms  and  a  Tread  between,  All 
Connected  with,  and  Dividing  Equally,  a  lo"  Cylinder. — This  plan  of  stairs  is  given  at 
Pl.\te  No.  5,  Fig.  9.  Describe  the  centre  line  of  rail,  and  space  the  balusters  as  required. 
Draw  the  tangents  A  B,  BE,  EG  and  G  J  to  the  centre  line  of  rail  A  E  J  To  find  the  angles 
of  inclination  of  these  tangents,  and  other  points  of  measurement  by  which  *o  get  the  lengths 
of  balusters,  it  is  necessary  to  set  up  an  elevation. 

Fig.  2.  Elevation  as  given  at  Plan  Fig.  i  and  as  Figured.— Measure  the  three  treads 
in  the  cylinder  on  the  centre  hne,  and,  as  before  explained,  of  treads  situated  .^n  curves.  Let 
the  bottom  line  ot  rail  pass  through  the  centres  of  short  balusters  at  X,  1  below  and  X  X 
above;  draw  the  centre  line  of  rail  LC  parallel  to  X,  1,  and  the  centre  line  FG  paiallel  to 
X  X;  make  A  B  and  J  G  equal  A  B  and  J  G  of  Fig.  i.  From  A  at  right  angks  to  the  chord- 
line  draw  A  B,  and  from  G  at  right  angles  to  the  chord-line  draw  G  J;  from  C  at  right 
angles  to  H  E  draw  C  E;  divide  E  J  at  D  in  two  equal  parts.  Place  the  centres  of  balusters 
1,  2,  3  as  at  the  plan,  and  draw  lines  through  each  parallel  to  the  rise-lines  and  indefinitely. 

At  Fig.  I,  to  further  Prepare  for  the  Development  of  the  Centre  Line  of  Wreath 
and  for  the  Lengths  of  Balusters: — Make  the  angles  A  C  B  and  J  H  G  equal  A  C  B  and  J  H  G 
of  Fig.  2;  prolong  J  G  to  F,  and  K  E  to  D;  make  E  D  and  G  F  each  equal  E  D  of  Fig.  2; 
connect  D  B;  connect  F  E;  make  G  U  equal  J  H;  draw  U  X  parallel  to  G  E;  make  X  W  parallel 
to  G  F;  connect  W  K,  which  is  the  directing  level  line  for  the  quarter-circle  E  J;  make  D  N 
equal  B  C;  draw  N  L  parallel  to  E  B,  and  LO  parallel  lo  E  D;  connect  0  K:  then  0  K  will 
be  tlie  directing  level  line  for  the  quarter-circle  A  E;  from  the  centre  of  baluster  1  draw  1  S 
parallel  to  K  0;  parallel  to  B  C  draw  ST;  from  the  centre  of  baluster  2  draw  2V  parallel 
to  K  0;  make  V  M  parallel  to  E  D;  through  the  centre  of  baluster  3  draw  3  Z  parallel  to  K  W; 
make  Z  Y  parallel  to  G  F.  At  Fig.  2  make  S  T  equal  S  T  of  Fig.  1,  and  V  M  equal  V  M  of 
Fig.  i;  make  ZY  equal  Z  Y  of  Fig.  i;  through  ATMYH  trace  the  centre  line  of  wreath, 
and  set  off  each  side  of  the  centre  half  the  thickness  of  rail  as  shown  by  the  dotted  lines.  Of 
the  three  balusters  around  this  cylinder,  only  one.  No.  3,  will  require  a  half  inch  more  length 
than  usual  for  short  balusters.  A  L  and  H  F  are  4"  straight  wood  to  be  left  on  the  wreath- 
pieces  above  and  below  the  cylinder. 

Fig.  3.  Plan  of  Rail  and  the  Centre  Line  of  the  First  Quarter  AE  with  the  Tangent 
A  B  and  B  E,  and  the  Angles  of  Inclination  ACS  and  B  D  E,  from  Fig.  i.— Make  D  N  equ£ 
BC;  draw  N  L  parallel  to  E  B,  and  LO  paralh-l  to  ED;  connect  OK,  the  directing  level  line 
parallel  to  0  K  draw  4,  I,  B  U,  W  X  and  AH;  parallel  to  B  C  draw  XG;  parallel  to  D  E  draw 
Y  M;  at  right  angles  to  0  K  draw  AQ  indefinitely;  and  again  at  right  angles  to  0  K  draw  E  R 
indefinitely;  on  B  as  centre  with  B  D  as  radius  describe  the  arc  D  R;  on  B  again  as  centre  with 
C  A  as  radius  describe  an  arc  at  Q;   connect  Q  R. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  E: — 
Prohmg  B  E  to  V;  make  EV  equal  N  P;  connect  V  K:  then  the  bevel  at  V  will  contain  the  angle 
required. 

To  Find  the  Angle  with  which  to  Square  the  Wreai;h-piece  at  the  Joint  over  A: — 

At  U  draw  U  T  at  right  angles  to  KA;  make  UT  equal  B  F;  connect  T  A;  then  the  bevel  at  T 
will  contain  the  angle  sought. 

Fig,  4.  Face-mould  from  Plan  Fig.  3;  also  Squaring  the  Wreath-piece  at  Both 
Joints. — Draw  the  line  Q  R;  make  RS  and  SQ  equal  the  same  at  Fig.  3.  On  Q  as  centre  with 
A  C  of  Fig.  3  as  radius  describe  an  arc  at  B;  on  R  as  centre  with  D  B  of  Fig.  3  as  radius  intersect 
the  arc  at  B;  from  S  test  the  intersections  at  B  by  S  B  of  Fig.  3;  connect  R  B,  Q  B  and  S  B; 
prolong  B  Q  to  C;  make  QC  equal  A  L  of  Fig.  2;  make  the  joint  C  and  R  at  right  angles  to 
the  tangents;  make  QG  equal  A  G  of  Fig.  3;  make  R  M  L  equal  D  M  L  of  Fig.  3;  parallel  to  S  B 
draw  Q  H,  G  W,  L  J  and  4,  I;  make  M  I  and  M  4  equal  Y  I  and  Y  4  of  Fig.  3;  make  L  J  and  B  Z  Y 
equal  0  J  and  B  Z  Y  of  Fig.  3;  make  G  W  and  Q  H  equal  X  W  and  A  H  of  Fig.  3;  through  R 
draw  4  P;  make  R  P  equal  R  4;  through  Q  draw  W  F;  make  Q  F  equal  Q  W;  from  F  and  W 
draw  lines  to  joint  C  parallel  to  B  C;  through  F  H  Z  I  P  of  the  convex  and  W  Y  J  4  of  the 
concave  trace  the  curved  edges  of  the  face-mould.  The  slide-line  is  drawn  at  right  angles  to 
the  level  directing  line  S  B.  Joint  R  of  the  wreath-piece  is  squared  by  the  bevel  at  V  of 
Fig.  3,  and  joint  C  by  the  bevel  at  T  of  Fig.  3.  The  face-mould  is  explained  in  detail  at  Plate 
No.  12.  The  development  of  th<  centre  line  of  the  wreath-piece  is  explained  at  Plate  No.  20: 
the  quarter-circle  Q  V  of  Fig.  3,   'ind  Q  Y  of  Fig.  4. 


Plate  No.  41 


Plate  No. 42 


PLATE  42. 

Fig.  I.  Plan  of  Landing  and  Starting  Two  Flights,  Both  in  Connection  with  a  12" 
Cylinder. — This  plan  is  given  at  Plate  No,  6,  Fig.  3.  The  centre  line  of  tlie  rail  may  be 
described,  the  balusters  spaced,  and  the  tangents  to  the  centre  line  drawn;  then,  to  find  the 
angle  of  inclination  of  these  tangents,  an  elevation  of  the  plan  must  be  made. 

Fig.  2.  Elevation  of  Plan  as  given  at  Fig.  i  and  as  Figured  ;  also  Development 
of  the  Centre  Line  of  Wreath. — In  setting  up  the  elevation,  the  treads  in  the  cylinder  must 
be  measured  between  chord-lines  on  the  centre  line  of  rail,  each  in  two  parts,  so  as  to  get 
practically  near  enough  to  the  stretch-out  of  the  circle.  Draw  the  chord-lines  in  the  position 
given  on  the  plan,  and  parallel  to  the  rise-lines.  Place  the  centre  of  balusters  as  given  on 
each  step  of  the  plan,  and  through  these  centres  draw  lines  parallel  to  the  rise-lines  indefinitely. 
Let  the  bottom  line  of  rail  above  and  below  each  pass  through  the  centres  of  short  balusters 
X  X;  draw  the  centre  line  of  rail  at  I  G  and  T  B  each  parallel  to  X  X.  Make  the  distance 
from  the  chord-line  AW  to  BC  equal  tiie  tangent  A  C  of  Fig.  i;  draw  BC  parallel  to  AW; 
at  A  draw  A  C  at  right  angles  to  A  W;  make  the  distance  from  the  chord-line  J  S  to  G  equal 
the  tangent  H  G  of  Fig.  i;  draw  G  F  parallel  to  J  S;  at  G  draw  G  H  at  right  angles  to  J  S; 
from  B  draw  B  F  parallel  to  the  line  of  floor;  divide  G  F  in  two  equal  parts  at  D;  transfer 
the  angle  ABC  to  ABC  of  Fig.  i.  The  three  remaining  tangents  all  happen  in  this  case  to 
have  the  same  angles  of  inclination,*  so  that  the  angle  H  J  G  may  be  placed  at  E  D  C,  G  F  E 
and  H  J  G,  Fig.  i.  Make  D  H  of  Fig.  i  equal  C  B;  draw  H  J  parallel  to  E  C;  make  J  V  parallel 
to  E  D;  connect  V  K,  which  is  the  directing  level  line  for  the  quarter  E  A.  The  directing 
level  line  for  the  quarter  E  H — that  quarter  having  a  common  angle  of  inclination — is  a  line 
drawn  from  G  to  K.  From  the  centie  of  baluster  1  draw  1  L  parallel  to  C  B;  from  the  centre 
of  baluster  2,  parallel    to   K  V,  draw  2,0;  parallel  to  E  D  draw  OK  and   3  M;    from  balusters 

4  and  5  draw  4N  and  5Q  parallel  to  KG;  parallel  to  G  F  draw  N  P;  jiarallel  to  H  J  draw 
Q  R.  At  Fig.  2  make  Z  R  over  baluster  5  equal  Q  R  of  Fig.  i.  Make  N  P  over  baluster  4 
equal  N  P  of  Fig.  i;  make  E  M  over  baluster  3  equal  3  M  of  Fig  i;  make  B  0  equal  0  K 
of  Fig.  i;  parallel  to  B  F  draw  0  K;  make  V  L  over  baluster  1  equal  1  L  at  Fig.  i;  through 
J  R  P  M  K  LA  trace  the  centre  line  of  wreath;  set  off  each  side  of  the  centre  half  the  thickness 
of  rail  as  shown  by  the  dotted  lines. 

To  Find  the  Lengths  of  Balusters: — For  example,  take  baluster  3;  3  U  measures  61", 
which  must  be  added  to  2'. 2",  the  length  of  short  balusters  X  X,  and  this  makes  the  length  of 
that  baluster,  measured  along  its  centre  from  top  of  step  to  under  side  of  rail,  2'.8f". 

Fig.  3.  Plan  of  Rail  for  Quarter-circle,  Centre  Line  A  E,  Fig.  i,  with  Angles  of 
Inclination— over  Tangents — A  B  C  and  C  D  E  ;  also  the  Directing  Level  Line  V  K  as 
Transferred  from  Fig.  i. — Parallel  to  K  V  d  raw  Z  5,  C  Q  and  Y  W;  parallel  to  E  D  draw  4  F; 
parallel  to  C  B  draw  XN;  at  right  angles  to  V  K  draw  ET  and  AO  indefinitely;  on  C  as 
centre  with  C  D  as  radius  describe  the  arc  D  T;  again  on  C  as  centre  with  B  A  as  radius 
describe  an  arc  at  0;  connect  0  T. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  E:  — 
Prolong  C  E  to  S;  make  E  S  equal  H  G;  connect  S  K:  then  the  bevel  at  S  contains  the  angle 
required. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  A;— 

Froin  Q  draw  Q  P  at  right  angles  to  K  A;  make  Q  P  equal  C  R;  connect  P  A:  then  the  bevel 
at  P  contains  the  angle  sougiit. 

Fig.  4.  Face-mould  from  Plan  Fig.  3;  also  Showing  the  Squaring  of  Wreath-piece 
at  Both  Joints. — Draw  the  line  A  T  indefinitely;  make  T  U  and  U  A  equal  T  U  and  U  0  ot 
Fk;.  3.  On  T  as  centre  with  C  D  of  Fig.  3  as  radius  describe  an  arc  at  B;  on  A  as  centre  with 
A  B  of  Fig.  3  as  radius  intersect  the  arc  at  B;  with  U  C  of  Fig.  3  test  the  intersection  of  the  arcs 
at  U  B;  connect  T  B,  B  A  and  U  B;  make  B  F  equal  C  F  of  Fig.  3;  make  B  N  equal  B  N  of  Fig.  3; 
parallel  to  U  B  through  F  and  N  draw  5  Z  and  W  Y;  make  F  Z  and  F  5  equal  4  Z  and  4,  5  of 
Fig.  3;  make  B  I  U  equal  C  I  U  of  Fig.  3;  make  N  Y  and  N  W  equal  X  Y,  X  W  of  Fig.  3;  through 
A  drawYC;  make  AC  equal  AY;  through  T  draw  ZG;  make  TG  equal  TZ;  make  A  J  for  straight 
wood  equal  A  T  of  Fig.  2;  make  the  joints  J  and  T  at  right  angles  to  the  tangents;  from  Y  and  C 
draw  lines  to  joint  J  parallel  to  B  J ;  through  G  5  I  W  C  of  the  convex  and  Z  U  Y  of  the  concave 
trace  the  curved  edges  of  the  face-mould.  The  wreath-piece  is  squared  at  joint  J  by  the  bevel 
at  P  of  Fig.  3,  and  joint  T  is  squared  by  the  bevel  at  S  of  Fig.  3.  The  slide-line  is  drawn  at 
right  angles  to  the  level  line  U  B.  A  face-mould  of  this  kind  is  explained  at  Plate  No.  12, 
and  the  development  of  the  centre  line  of  such  a  wreath-piece  as  this  is  given  at  Plate  No.  20; 
quarter-circle  Q  V  of  Fig.  3,  and  Q  Y  of  Fig.  4. 

Fig.  5.  Plan  of  Rail  for  Quarter-circle  E  H  of  Fig.  i,  with  Angles  of  Inclination  over 
Tangents  E  F  G  and  G  J  H,  as  Transferred  from  E  H  of  Fig.  i. — Connect  G  K;  parallel  to 
G  K  draw  L  Y;  parallel  to  G  F  draw  Z  D;  through  H  and  E  draw  H  W  indefinitely;  on  G  as  centre 
with  E  F  as  radius  describe  an  arc  at  W. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  Both  Joints : — Prolong 
G  H  to  N;  make  H  N  equal  H  R;  connect  N  K:  then  the  bevel  at  N  will  contain  the  angle 
required. 

Fig.  6.  Face-mould  from  Plan  Fig.  5 ;  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints. — Draw  the  line  W,  W;  make  S  W,  S  W  each  equal  S  W  of  Fig.  5;  draw 

5  G  at  right  angles  to  S  W;  make  S  G  equal  S  G  of  Fig.  5;  connect  G  W  and  G  W;  prolong 
G  W  to  A;  make  W  A  for  straight  wood  equal  J  I  of  Fig.  2;  make  the  joints  A  and  W  at  right 
angles  to  the  tangents;  make  G  D,  G  D  each  equal  F  D  of  Fig.  5;  through  D  and  D  draw  L  Y  and 
L  Y  parallel  to  G  S;  make  D  Y  and  D  L  at  both  sides  of  the  centre  each  equal  Z  Y  and  Z  L 
of  Fig.  5;  through  W  and  W  draw  L  M  and  L  M;  make  W  M  equal  W  L;  make  G  T  equal 
G  T  of  Fig.  5;  through  M  Y  T  Y  M  of  the  convex  and  L  S  L  of  the  concave  trace  the  curved 
edges  of  the  face-mould;  from  L  and  M  draw  lines  to  joint  A  parallel  to  G  A.  A  face-mould 
of  this  kind  is  explained  in  detail  at  Plate  No.  ii,  and  the  development  of  the  centre  line 
of  wreath-piece  at  Plate  No.  20,  Fig.  i,  quarter-circle  A  C,  and  Fig.  2,  A  E. 

*  //  the  height  C  F  had  been  more  or  less  than  twice  the  height  H  J,  then  the  fuce-mtuld  for  the  quarter-circle  H  E  would 
have  been  of  tlie  same  kind  as  that  of  FiG.  4. 


PLATE  43 


Fig.  I.  Plan  of  the  Starting  of  a  Staircase  with  One  Parallel  Step  at  the  Centre  of 
a  15"  Cylinder,  together  with  a  Quarter  Platform. — This  plan  is  given  at  Plate  No.  7, 
Fig.  9.  The  centre  line  of  rail  may  be  described,  the  tangents  drawn,  and  the  balusters 
spaced  as  required;  but  to  find  the  angle  of  inclination  over  the  plan  tangents,  etc.,  the  ele- 
vation must  first  be  set  up. 

Fig.  2.    Elevation  of  Treads  and  Rises  as  given  at  Plan  Fig.  i,  and  as  Figured. — 
Let  the  bottom  line  of  rail  be  drawn  through  the  centres  of  short  balusters  X  X,  and  draw  the 
centre  line  of  rail  A  N  parallel  to  X  X.    Place   the  chord-line  H  J  parallel  to   the  rise-line,  and 
in  position  as  at  the  plan;  make  J  G  equal  J  F  at  Fig.  i;  draw  F  G  parallel  to  the  rise-line, 
and  at  the    intersection  G   draw  G  J  at  right   angles  to  the    rise-lines.     Make  F  D  equal  F  E 
of  Fig.  i;  draw  E  D  parallel  to  the  rise-line;  at   the  intersection  D  draw  D  F  at   right  angles 
to  the  rise-line;  make  EC  equal  E  C  of  Fig.  i;  make  Z  1  equal  Zl  of  Fig.  i;  make  1  B  equal  1  B  of 
Fig.  i;  draw  B  N  at  right  angles  to  the  line  of  floor;  make  BK  equal  4",  and  KN  half  the  thickness 
of  rail;   draw  N  E  parallel  to  the  floor-line.      Place  the  centre  of  each  baluster  on  the  treads  as 
numbered,  and  as  fixed  around  the  centre  line  of  rail  at  the  plan,  and  draw  lines  through  each 
parallel  to    the    rise-lines    indefinitely.    At  Fig.  i    make  the  angles    of   inclination  J  H  F,  F  G  E 
and  EDO  all  equal  J  H  G  of  Fig.  2;  parallel  to  A  D  through  the  centre  of  baluster  2  draw  N  M; 
from  the  centre  of   baluster    1   draw  1  L  parallel    to  A  D;  draw  I  K    parallel  to  A  D.  Connect 
F  A;  from  3  parallel  to  A  F  draw  3,  0;  parallel  to    F  G  draw  0  R;  parallel  to    A  F  draw  4  Q; 
parallel  to  J  H   draw  Q  P.    Again  at  Fig.  2,  baluster  1,  make  V  L  equal  V  L  of  FiG.  i;  at  bal- 
uster   2  make    S  M  equal    S  M  of  Fig.  i;  at    baluster  3  make  0  R    equal    0  R  of   Fig.   i;  at 
baluster  4  make  Q  P  equal  Q  P  of  Fig.   i.    Through  the  points  N  L  M  R  P  H  trace  the  centre 
line  of  wi  eath-piece;  set  off    half    the  thickness  of   rail  each  side  of    the  centre  as  shown  by 
the  dotted  lines.    To  find  the  length  of  any  of  these  balusters  proceed  as  before  directed. 

Fig.  3.  Face-mould  from  Plan  Fig.  i,  Quarter-circle  B  E,  also  Showing  the  Squaring 
of  the  Wreath-piece  at  Both  Joints. — Draw  the  lines  D  C  and  C  H  at  right  angles.  Make 
C  D  equal  C  D  of  Fig.  i;  make  C  B  equal  C  B  of  Fig.  i;  make  B  H  equal  3"  for  straight 
wood.  Make  C  K  M  equal  C  K  M  of  Fig.  i;  draw  K  I  parallel  to  C  B;  draw  Q  N  through  M 
parallel  to  C  B;  draw  the  joint  D  parallel  to  C  B;  draw  I  T  parallel  to  D  C;  make  the  joint 
at  H  at  right  angles  to  C  H;  from  T  and  I  parallel  to  C  H  draw  lines  to  the  joint  H;  make 
C  A  and  K  J  each  equal  C  H  and  T  U  of  Fig.  i;  make  M  N  and  M  Q  each  equal  S  N  and 
S  W  of  Fig.  i;  make  D  Y  and  D  X  each  equal  E  Y  of  Fig.  i.  Through  T  A  J  Q  X  of  the  con- 
vex and  I  N  Y  of  the  concave  trace  the  curved  edges  of  the  face-mould.  The  angle  with  ivhich 
to  square  the  jvrcath  piece  at  Joint  H  is  contained  in  the  bevel  at  D  of  FiG.  i.  The  sides  of  the 
wreath-piece  at  joint  D  are  at  right  angles  to  the  face  of  plank,  and  the  over-wood  is  taken 
off  both  surfaces  of  the  plank  equally.  Hand-rails  much  thicker  than  two  thirds  of  their  width 
require  considerably  greater  width  and  thickness  of  stuff  to  work  out  the  wreath-piece. 

Fig.  4.  Plan  of  Quarter-circle  taken  from  J  E  of  Fig.  i,  together  with  Angles  of 
Inclination  over  the  Plan  Tangents  Lettered  Alike. — Connect  F  K;  through  E  and  J  draw 
E  L  indefinitely;  on  F  as  centre  with  F  H  as  radius  describe  the  arc  H  L.  Parallel  to  F  K 
draw  Y  D  and  V  B;  parallel  to  F  G  draw  Z  X  and   N  C. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  Both  Joints : — Pro- 
long F  J  to  P  indefinitely;  make  J  P  equal  J  M;  connect  P  K:  then  the  bevel  at  P  will  con- 
tain the  angle  required. 

Fig.  5.  Face-mould  from  Plan  Fig.  4;  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints,  and  the  Additional  ^Vidth  Required  by  a  Form  of  Hand-rail  of  this 
Proportion. — Draw  the  line  E  H  indefinitely;  make  W  H  and  W  E  each  equal  W  L  of  Fig.  4.  At 
right  angles  to  E  H  draw  WG;  make  W  G  equal  W  F  of  Fig.  4;  connect  G  E,  G  H  and  G  W;  make 
G  C  X,  G  C  X  equal  the  same  of  Fig.  4.  Through  G  C  X  each  side  of  the  centre  parallel  to  G  W 
draw  D  Y,  B  V;  make  X  Y  and  X  D  at  each  end  equal  Z  Y  and  Z  D  of  Fig.  4;  make  C  K  V  and 
C  B  each  side  of  the  centre  equal  N  K  V  and  N  B  of  Fig.  4;  make  G  T  S  U  equal  F  T  S  U  of 
Fig.  4;  through  H  and  through  E  draw  Y  0;  make  E  0  equal  E  Y;  make  H  0  equal  H  Y; 
make  H  A  equal  H  A  of  Fig.  2;  from  0  and  Y  draw  lines  to  joint  A  parallel  to  G  A;  through 
ODBTBDO  of  the  convex  and  through  Y  V  U  V  Y  of  the  concave  trace  the  curved  edges 
of  the  face-mould.  The  joints  A  and  E  of  the  wreath-piece  are  squared  by  the  angle  con- 
tained in  the  bevel  at  P  of  FiG.  4.  By  laying  out  the  squaring  of  the  joint  as  here  given, 
the  thickness  and  width  of  wood  required  to  work  out  the  wreath-piece  are  found,  and  with 
half  this  width  as  radius  on  each  of  the  centres  E  K  S  K  H  A  describe  arcs  of  circles,  touch- 
ing which  the  edges  of  a  parallel  pattern  may  be  traced;  or  no  pattern  need  be  made,  and 
the  increased  width  of  wood  required  may  be  scribed  from  the  edges  of  the  face-mould  on 
the  plank.  The  development  of  a  centre  line  of  wreath  is  given  in  detail  at  Plate  No.  20,  FiGS. 
I  and  2.  Face-mould  FiG.  3  is  given  in  detail  at  Plate  No.  10.  Face-mould  Fig.  5  is  given  in 
detail  at  Plate  No.  ii. 


Plate  No.  43 


G 


PLATE  44. 

Fig.  I.  A  Superior  Plan  of  Starting  a  Stairs  making  a  Quarter-turn,  with  Parallel 
Steps  and  Platform,  in  about  the  Same  Space  Required  when  Planned  with  Winders. — 

This  plan  is  given  at  Plate  No.  7,  Fig.  8.  Describe  the  plan  of  rail  and  its  centre  line;  draw 
the  tangents  D  B  and  B  A,  and  space  the  balusters  as  required;  then  before  proceeding  further 
the  elevation  must  be  set  up. 

Fig.  2.  Elevation  of  Treads  and  Risers  as  given  at  Plan  Fig.  i  and  as  Figured; 
also  the  Development  of  the  Centre  Line  of  the  Wreath. — Draw  the  treads  and  rises  as 
shown  on  the  plan,  taking  the  measure — on  the  centre  line  of  the  rail — of  each  tread  in  two 
parts  as  before  explained.  Place  the  chord-line  A  F  as  at  S  on  the  plan  Fig.  i.  Let  the 
bottom  line  of  rail  rest  on  X  X,  the  centres  of  short  balusters;  draw  the  centre  line  of  rail 
TE  indefinitely  and  parallel  to  XX;  make  FE  equal  D  B  of  Fig.  i;  draw  EG  parallel  to  the 
chord-line;  make  1  S  equal  four  inches,  and  S  B  half  the  thickness  of  rail;  parallel  to  the  floor- 
line  draw  B  D;  make  D  B  equal  G  C:  then  B  is  a  fixed  point,  and  E  is  also  a  fixed  point 
at  the  place  where  the  line  G  E  intersects  the  centre  line  of  rail  T  E.  Connect  E  B;  and 
where  the  line  C  D  intersects  the  line  E  B  at  C,  draw  the  line  C  G  at  right  angles  to 
CD.  1  Z  equals  1  A  of  Fig.  i;  make  Z  V  parallel  to  1  B.  Place  the  centre  of  each  baluster 
on  the  steps  in  position,  and  number  them  as  at  the  plan,  drawing  lines  through  each  par- 
allel to  the  rise-lines  indefinitely.  Let  A  T  be  three  inches  for  straight  wood  to  be  put  on 
that  end  of  the  wreath-piece.  At  Fig.  i  make  the  angle  D  C  B  equal  D  C  B  of  Fig.  2; 
parallel  to  E  C  through  3  draw   N  G,  and  from   H   draw  H  F  parallel  to  A  B. 

Fig.  3.  Plan  of  Rail  Quarter-circle  D  S  of  Fig.  i,  with  Centres  of  Balusters  4,  5 
and  6  in  Place  on  the  Centre  Line. — Make  the  angle  F  A  E  equal  F  A  E  of  Fig.  2;  make  the 
angle  E  H  J  equal  G  E  C  of  Fig.  2;  make  E  P  equal  F  A;  draw  P  Q  parallel  to  E  J;  draw  Q  T 
parallel  to  H  E;  connect  T  N,  the  directing  level  line;  parallel  to  T  N  draw  X  I,  E  Y,  5,  2  and 
W  V;  parallel  to  E  H  draw  2  R  and  4  U;  parallel  to  F  A  draw  0  C.  At  Fig.  2  baluster  1  hap- 
pens to  be  at  the  point  B;  at  baluster  2  make  N  K  equal  I  F  of  Fig.  i;  at  baluster  3  make 
J  H  equal  J  G  of  Fig.  i;  at  baluster  4  make  L  U  equal  4  U  of  Fig.  3;  at  baluster  5  make 
P  R  equal  2  R  of  Fig.  3;  at  baluster  6  make  0  Q  equal  0  C  of  Fig.  3;  through  V  K  H  U  R  Q  A 
trace  the  centre  line  of  the  wreath.  Set  off  each  side  of  the  centre  half  the  thickness  of  rail 
as  shown  by  the  dotted  lines.  To  find  the  length  of  balusters  proceed  as  before  directed. 
Again,  at  Fig.  3,  parallel  to  F  E  draw  C  G;  at  right  angles  to  T  N  draw  F  L  and  J  K;  on  E 
as  centre  with  E  A  as  radius  describe  the  arc  A  L;  again,  on  E  as  centre  with  H  J  as  radius 
describe  an  arc  at  K;  connect   K  L 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  J: — 
Prolong  FN  to  M;  make  N  M  equal  T  S;  connect  M  J;  then  the  bevel  at  M  contains  the  angle 
sought. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  over  Joint  F  : — Prolong 
E  F  to  D;  make  F  D  equal  G  B;  connect   D  X;  then  the  bevel  at   D  contains  the  angle  required. 

Fig.  4.  Face-mould  from-  Plan  Fig.  3,  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints. — Draw  the  line  A  J  indefinitely;  make  YA  and  YJ  equal  Y  L  and  Y  K 
of  Fig.  3;  on  J  as  centre  with  J  H  of  Fig.  3  as  radius  describe  an  arc  at  H;  on  A  as  centre 
with  A  E  of  Fig.  3  as  radius  intersect  the  arc  at  H;  on  Y  as  centre  with  Y  E  of  Fig.  3  as  radius 
test  the  intersection  of  the  arcs  at  H;*  connect  J  H,  H  A  and  Y  H;  make  H  Q  U  equal  H  Q  U 
of  Fig.  3;  make  H  C  equal  E  C  of  Fig.  3;  parallel  to  Y  H  through  C  draw  X  I;  through  Q 
and  U  parallel  to  Y  H  draw  Q  8  and  V  W;  make  C  X  and  CI  equal  0  X  and  0  I  of  Fig.  3; 
make  H  Z  9  equal  E  Z  9  of  Fig.  3;  make  Q  8  equal  T  8  of  Fig.  3;  make  U  W  and  U  V  equal 
4  W  and  4  V  of  Fig.  3.  Through  J  draw  W  B;  make  J  B  equal  J  W;  through  A  draw  X  D; 
make  A  D  equal  A  X;  make  A  K  equal  T  A  of  Fig.  2;  make  the  joints  K  and  J  at  right  angles 
to  the  tangents;  from  X  and  D  draw  lines  to  joint  K  parallel  to  H  K;  through  B  V  Z  I  D  of 
the  convex  and  W  8,  9,  X  of  the  concave  trace  the  curved  edges  of  the  face-mould.  The  slide- 
line  is  made  at  right  angles  to  the  level  line  Y  H.  The  angle  zmth  which  to  square  the  ivreath-piece 
at  joint  K  is  taken  by  the  bevel  D  of  Fig.  3,  and  for  joint  J  the  bevel  M  of  Fig.  3. 

Fig.  5.  Face-mould  from  Plan  Fig.  i.  Quarter  A  D,  which  Joins  the  Level  Rail;  also 
Showing  the  Squaring  of  the  Wreath-piece  at  the  Joints. — Draw  the  lines  B  C  and  B  Z 
at  right  angles;  make  B  F  G  C  equal  the  same  at  Fig.  i;  make  B  A  equal  B  A  of  Fig.  i; 
through  A  draw  H  E  indefinitely  and  at  right  angles  to  B  A;  parallel  to  B  A  draw  F  H,  and 
through  G  and  C,  M  N  and  0  0;  make  F  L  equal  I  L  of  Fig.  i;  make  G  N  and  G  M  equal 
J  N  and  J  M  of  Fig.  i.  Joint  C  is  made  at  right  angles  to  B  C;  make  C  0  and  C  0  each 
equal  D  0  of  FiG.  i;  make  A  Z  three  inches  for  straight  wood;  make  the  joint  Z  at  right 
angles  to  B  Z;  make  A  E  equal  A  H;  draw  lines  from  H  and  E  to  joint  Z  parallel  to  B  Z. 
Through  0  M  L  K  E  of  the  convex  and  0  N  H  of  the  concave  trace  the  curved  edges  of  the 
face-mould.  The  sides  of  the  wreath  at  joint  C  are  made  at  right  angles  to  the  face  of  the 
plank,  f  The  angle  with  which  to  square  the  wreath  at  joint  Z  is  taken  by  the  bevel  C  of 
Fig.  I.  Face-mould  Fig.  4  is  explained  in  detail  at  Plate  No.  12.  Face-mould  Fig.  5  is 
explained  in  detail  at  Plate  No.  10.  The  development  of  a  centre  line  of  wreath  as  at  Fig.  2  is 
given  in  detail  at  Plate  No.  20,  Figs.  3  and  4. 

*  In  face-moulds  of  this  kind,  if  the  drawing  is  carefclly  made,  instead  of  the  length  of  this  level  line,  Y  H  being 
applied  as  a  test,  it  ?nay  be  used  mith   the  length  of  either  one  of  the  tangents  to  establish  the  angular  point  H. 

f  The  sides  of  wreaths  are  straight  vertically,  and  can  be  worked  correctly  only  in  that  direction  with  suitable  hol- 
lows and  rounds.  After  a  wreath  is  shaped,  if  a  straight-edge  is  tried  square  across  the  side  of  it  at  any  point  it 
will  be  found  hollow  on  the  concave  side  and  rounding  on  the  convex  side.  For  this  reason  in  cases  like  face-mould 
Fig.  5,  joint  C — in  fact,  at  centre  joints  of  all  face-moulds — should  properly  have  some  over-wood  at  O  O  left  on  in 
marking  out  the  stuff. 


PLATE  45 


Fig.  I.    Plan  of  a  Circular  Flight  of  Stairs  Winding  Around  a  Circular  Post. — In 

planning  these  stairs  the  first  tiling  tu  do  is  to  fix  tlie  place  of  the  starting  riser.  The  next 
consideration  is  the  landing  and  head-room.  In  this  case  the  first,  or  starting,  rise  becomes 
the  landing  and  seventeenth  rise,  making  one  revokition;  the  whole  height — the  rises  being  8" — 
is  therefore  ii'.4.".  In  finding  the  head-room  at  a  point  between  the  starting  and  landing  rises, 
from  the  landing  deduct  lo"  floor-beam,  and  2"  for  floor  and  plaster;  deduct  also  the  bottom 
rise,  8" — all  together  20",  to  be  taken  from  11'. 4",  leaving  g'.8"  head-room.  Then,  again,  the  floor 
at  the  landing  is  brought  on  a  line  of  the  fifth  rise  E  B,  which  makes  it  necessary  to  deduct 
from  the  last  sum  four  rises, — 32", — leaving  a  balance  of  head-room  at  that  point  of  y'.o".  The 
post  is  sometimes  cased  as  shown.  The  tread  around  the  line  of  travel  is  7",  about  the  least 
that  ought  to  be  permitted,  if  a  hand-rail  *  is  put  around  the  post;  but  if  a  rail  is  hung  over 
the  outside  string,  then  the  line  of  travel  would  be  further  out,  and  give  a  tread  of  9I",  which 
would  leave  the  plan  as  it  is,  ample.  If  the  staircase  is  to  stand  independent  of  wall  or 
partition,  the  string  should  be  bent  laminated — see  Plate  No.  8,  Fig.  5 — as  being  stronger  than 
by  any  other  method.  To  give  support  to  the  string  it  may  be  enclosed  to  the  floor  from  A  to  C, 
or  from  A  to  B,  and  then  set  a  supporting  post  at  C,  and  at  D  suspend  the  string  with  a  small 
iron  rod  from  the  floor-beam  of  the  story  above.  If  the  circular  post  is  made  large  enough,  each 
riser  can  be  set  into  mortices  in  the  post  four  or  five  inches,  and  secured  with  lag-screws;  the 
steps,  too,  sliould  be  let  into  the  post  about  2". 

Figs.  2,  3  and  4. — Fig.  2  is  a  Plan  of  Stairs  such  as  is  given  at  Plate  No.  7,  Fig.  8, 
in  which  the  Cylinder,  5"  Thick,  is  Best  Built  Solid  and  Veneered,  Both  Faces  for  a 
Close  String. — The  manner  of  building  tiie  solid  cylinder  is  shown  by  the  two  thicknesses  of 
staves  between  tiie  veneers;  the  straight  portion,  built  with  the  cylinder  from  F  to  N,  is  so  worked 
for  the  purpose  of  including  the  easement  at  the  lower  edge  of  the  concave  face  Fig.  4. 

Fig.  3.  TM  veneer  laid  out  for  the  convex  face  of  the  cylinder,  the  lettering  agreeing  with  tliat 
face  on  the  plan  Fig.  2.  Tiie  vertical  and  irregular  lines  are  to  show  the  position  and  lengths 
of  staves  required  for  the  convex  face. 

Fig.  4.  The  veneer  laid  out  for  the  concave  face  of  the  cylinder,  the  lettering  agreeing  with  that 
face  on  the  plan  Fig.  2.  The  vertical  and  irregular  lines  indicate  the  lengths  and  position  of 
the  staves  for  the  concave  face  of  the  cylinder.  This  veneer  is  first  laid  over  the  prepared, 
rough-staved  cylinder,  and  on  the  veneer  the  concave-faced  staves  are  fitted  and  glued;  then  the 
convex  staves  are  fitted  and  glued  over  these  again;  and,  finally,  the  convex  veneer  Fig.  3  is 
glued  over  the  whole. 

Fig.  5.  Plan  of  Quarter  Platform  Stairs,  Showing  another  Way  of  Placing  the  Risers 
Connecting  with  the  Quarter  Cylinder. — A  riser  may  be  placed  at  the  chord  A;  then  taking 
A  B — which  is  the  tangent  to  the  centre  line  of  rail — and  a  portion  of  the  other  tangent,  B  D, 
to  equal  together  one  tread,  as  follows:  A  B  =  6^",  B  C  =  3^",  making  10"  one  tread  to  the  place 
of  the  next  riser  C. 

See  Plate  No.  5,  Fig.  10,  and  Plate  No.  37,  Fig.  5. 


*  For  treatment  of  liand-rail  over  circular  stairs  see  Plates  53  and  54. 


Plate  No. 45 


PLATE  46. 


Fig.  I.  A  Superior  Plan  of  Stairs  Making  a  Quarter-turn  at  the  Landing,  with 
Parallel  Steps  and  Platform,  in  about  the  Same  Space  required  when  Planned  with 
Winders.— This  plan  is  given  at  Plate  No.  7,  Fig.  4.  Describe  the  centre  Hne  of  rail,  and 
draw  the  tangents  at  each  quarter  cylinder.  Space  the  balusters  as  required.  To  find  the 
angles  of  inclination  over  the  plan  tangents,  and  other  measurements,  the  elevation  must  first 
be  set  up. 

Fig.  2.  Elevation  of  Treads  and  Rises  as  given  at  Fig.  i  on  the  Centre  Line  of 
the  Hand-rail ;  also  the  Development  of  the  Centre  Line  of  Wreath-piece. — Place  the 
ciiord-lines — of  which  there  are  three  needed — as  at  the  plan.  Put  the  centres  of  balusters  1, 
2,  3  as  given  on  each  tread,  and  draw  lines  indefinitely  through  these  centres  parallel  to  the 
rise-lines.  From  chord-line  A  make  A  B  equal  A  B  of  Fig.  i.  Through  B  draw  B  C  parallel 
to  the  rise-lines;  on  the  line  B  C — assuming  any  point  C  that  will  raise  the  wreath  a  little 
high  rather  than  fix  it  low — draw  C  F,  the  centre  line  of  ramp,  and  the  inclined  line  over  the 
first  plan  tangent;  where  this  line  C  F  intersects  the  chord-line  at  A  draw  A  B  at  right  angles 
to  the  chord-line;  make  E  Y  equal  E  B  of  Fig.  i:  then  Y  becomes  a  fixed  point.  Make  H  Z 
equal  P  V  of  Fig.  i;  draw  Z  Q  at  right  angles  to  HZ;  make  Z  0  four  inches,  and  OQ  half 
the  thickness  of  rail:  then  Q  is  a  fixed  point  (unalterable  if  the  level  rail  is  to  be  kept  at 
its  usual  height).  Connect  Q  Y;  divide  P  D  in  two  equal  parts  at  G;  make  A  F  four  inches 
for  straight  wood  to  be  left  at  the  lower  end  of  the  wreath-piece.  Again  at  Fig.  i  make  the 
angle  V  Q  P  equal  V  Q  P  of  Fig.  2;  draw  T  R  and  W  N  parallel  to  U  Q;  make  the  angle  E  D  B 
equal  E  D  Y  of  Fig.  2;  make  the  angle  B  C  A  equal  B  C  A  of  Fig.  2;  make  B  H  equal  E  D; 
parallel  to  B  A  draw  H  Z;  parallel  to  C  B  draw  Z  G;  connect  G  F,  the  directing  level  line; 
parallel  to  G  F  draw  2  M  and  1  K;  parallel  to  E  D  draw  M  L;  parallel  to  B  C  draw  K  J. 
Again  at  Fig.  2,  baluster  1,  make  K  J  equal  K  J  of  FiG.  i;  at  baluster  2  make  M  L  equal  M  L 
of  Fig.  i;  through  the  points  D  LJ  A  trace  the  centre  line  of  wreath-piece;  set  off  half  the 
thickness  of  rail  each  side  of  the  centre,  as  shown  by  the  dotted  lines. 

To  Find  the  Length  of  Balusters: — Take  for  example  baluster  2:  2  N  equals  which 
added  to  2'.2",  the  usual  length  of  short  baluster  at  X  (between  the  top  of  step  and  the 
bottom  of  rail),  makes  the  length  of  baluster  2  between  the  same  points  2'.^i". 

Fig.  3.  Face-mould  from  Plan  of  Rail  at  the  Landing  Quarter-circle  Fig.  i :  also 
Showing  the  Squaring  of  the  Piece  at  the  Joints. — Draw  the  lines  G  Q  and  Q  J  at  right 
angles;  make  Q  R  N  P  equal  Q  R  N  P  of  Fig.  i;  make  P  G  equal  P  G  of  Fig.  2;  make  Q  U 
equal  V  U  of  Fig.  i;  make  U  J  equal  2"  for  straight  wood;  make  joint  G  parallel  to  J  Q; 
make  T  K  and  joint  J  parallel  to  Q  G ;  parallel  to  J  Q  draw  1  T,  N  W  and  X  P  X;  make  P  X 
and  P  X  each  equal  P  X  of  Fig.  i;  make  R  I  and  Q  Y  equal  S  I  and  V  Y  of  Fig.  i;  draw  lines 
from  X  and  X  to  joint  G  parallel  to  P  G;  make  U  K  equal  U  T;  from  K  and  T  draw  lines 
to  joint  J  parallel  to  J  Q;  through  K  Y  I  X  of  the  convex  and  T  W  X  of  the  concave  trace 
the  curved  edges  of  the  face-mould.  The  angle  with  which  to  square  the  wreath  at  joint  J  is 
taken  by  the  bevel  Q  of  Fig.  i.  The  sides  of  wreath-piece  at  joint  G  are  made  at  right  angles 
to  the  face  of  the  plank,  and  the  over-wood  is  removed  equally  from  both  faces  of  the  plank. 
The  dotted  lines  show  the  increased  width  of  stuff  required  to  work  out  the  wreath-piece  of 
a  hand-rail  of  so  much  greater  depth  than  width. 

Fig.  4.  Plan  of  Hand-rail,  Quarter-circle  A  E  of  Fig.  i,  with  its  Angles  of  Inclination 
E  D  B  and  B  C  A.— Let  B  H  equal  E  D;  make  H  Z  parallel  to  B  A,  and  Z  G  parallel  to  C  B; 
connect  G  F,  the  directing  level  line;  parallel  to  G  F  draw  V  S,  B  5  and  R  I;  at  right  angles 
to  G  F  draw  A  P  and  E  W,  each  indefinitely;  on  B  as  centre  with  B  D  as  radius  describe  the 
arc  D  W;   again  on   B  as  centre  with  C  A  as  radius  describe  an  arc  at  P;   connect  P  W. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  E: — 
Prolong  B  E  to  T;  make  ET  equal  E  M;  connect  T  5:  then  the  bevel  at  T  will  contain  the 
angle  required. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  A: — 

Prolong  E  F  to  U;  make  F  U  equal  G  N;  connect  U  A:  then  the  bevel  at  U  will  contain  the 
angle  sought. 

Fig.  5.  Face-mould  from  Plan  of  Hand-rail  Fig.  4;  also  Showing  the  Squaring  ot 
the  Wreath-piece  at  the  Joints. — Draw  the  line  A  D  indefinitely;  make  X  D  and  X  A  equal 
X  W  and  X  P  of  Fig.  4.  On  A  as  centre  with  A  C  of  Fig.  4  as  radius  describe  an  arc  at  B; 
on  X  as  centre  with  the  level  line  X  B  of  Fig.  4  as  radius  intersect  the  arc  at  B;  connect 
A  B,  D  B  and  X  B;  prolong  B  D  to  G,  and  B  A  to  F;  make  A  F  equal  A  F  of  Fig.  2;  make  D  G 
equal  D  G  of  Fig.  2;  make  the  joints  G  and  F  at  right  angles  to  the  tangents;  make  D  K 
equal  D  K  of  Fig.  4;  make  B  Z  4  equal  C  Z  4  of  Fig.  4;  through  K  Z  4  parallel  to  B  X  draw 
SV,  LJ  and  IR  indefinitely;  make  K  V,  K  S  equal  Q  S,  Q  V  of  Fig.  4;  make  B  0,  X  0 
equal  the  same  at  Fig.  4;  make  Z  L  and  Z  J,  4,  I  and  4  R  equal  G  J  and  G  L,  Y  I  and  Y  R  of 
Fig.  4;  through  D  draw  V  C;  make  D  C  equal  D  V;  through  A  draw  R  H;  make  A  H  equal  A  R; 
from  R  and  H  draw  lines  to  joint  F  parallel  to  B  F;  from  V  and  C  draw  lines  to  joint  G 
parallel  to  B  G;  through  C  S  0  L  I  H  of  the  convex  and  V  0  J  R  of  the  concave  trace  the  curved 
edges  of  the  face-mould.  Four  additional  points  on  the  centre  line  may  be  measured  as  shown, 
and  a  parallel  pattern  made  by  describing  arcs  of  circles  of  a  radius  to  suit  the  width  given 
by  the  squaring  of  the  joints;  or  the  extra  width  may  be  scribed  on  the  plank  from  the  edges 
of  the  face-mould.  The  slide-line  is  drawn  at  right  angles  to  the  directing  level  line  B  X.  The 
angle  used  for  squaring  the  joint  G  is  taken  from  T  of  FiG.  4,  and  for  squaring  the  joint  F  the 
angle  at  the  bevel  U  of  Fig.  4.  J^or  detailed  explanation  of  face-mould  FiG.  3  see  Plate  No.  10 
and  for  face-mould  fiG.  5  see  Plate  No.  12.    See  Plate  No.  75. 


PLATE  47 


Fig.  I.  Plan  of  a  Part  of  a  Flight  of  Winding  Stairs  of  which  this  Portion  makes  a 
Half-turn. — This  plan  is  given  complete  at  Plate  No.  7,  Fig.  3.  Draw  the  centre  line  of  rail 
and  the  tangents.  Space  the  balusters  as  required;  then  to  find  angles,  measurements,  etc., 
proceed  to  set  up  the  elevation. 

Fig.  2.  Elevation  of  Treads  and  Rises  as  given  at  Plan  Fig.  i;  also  the  Develop- 
ment of  the  Centre  Line  of  the  Wreath. — Place  the  chord-lines — of  which  there  are  two,  A 
and  G — as  given  at  the  plan,  and  extend  the  upper  one,  J  D,  at  an  indefinite  length.  Put  the 
centres  of  the  balusters  on  each  step  as  at  the  plan,  and  draw  lines  through  these  parallel 
to  the  rise-lines  indefinitely.  With  a  straight-edge  held  in  the  direction  A  J  mark  a  point  at 
the  lower  chord,  A,  and  at  the  upper  chord,  J;  with  the  understanding  that  if  J  is  placed 
lower  on  the  chord-line,  then  the  upper  ramp  will  be  lengthened  and  the  wreath  brought 
lower;  and  if  A  is  raised  higher  on  the  chord-line,  then  the  lower  ramp,  already  about  right, 
would  be  increased  in  length  and  curve.  So  at  pleasure  fix  A  and  J;  then  draw  A  D  at  right 
angles  to  the  chord-line;  divide  D  J  in  four  equal  parts;  make  A  B  equal  the  tangent  A  B  of 
Fig.  1;  draw  B  C  parallel  to  the  rise;  make  B  C  equal  D  E;  connect  C  A,  and  prolong  to  N 
indefinitely.  At  right  angles  to  J  F  draw  G  H;  let  G  H  equal  G  H  of  FiG.  i;  connect  H  J,  and 
prolong  to  0  indefinitely;  make  J  0  and  A  N  each  3"  for  straight  wood  at  those  ends  of  the 
wreath-pieces.  Make  the  joints  of  ramps  at  N  and  0  at  right  angles  to  N  C  and  H  0.  Again 
at  FiG.  I,  as  the  four  heights  to  be  raised  at  each  tangent  are  alike,  make  all  the  angles  of 
inclination  BCA,  EFB,  H  KE  and  G  H  J  equal  B  C  A  of  Fig.  2.  Draw  the  directing  level 
lines  M  B  and  M  H;  parallel  to  M  B  draw  1  R  and  2  S;  parallel  to  B  C  draw  R  L;  parallel  to 
E  F  draw  ST.  Again  at  Fig.  2,  baluster  1,  make  R  L  equal  R  L  of  Fig.  i;  at  baluster  2, 
make  2  T  equal  S  T  of  Fig.  i;  at  baluster  3,  P  K  equals  H  K  of  Fig.  i.  Through  A  L  T  K  J 
trace  the  centre  line  of  the  wreath.  Set  off  each  side  of  the  centre  half  the  thickness  of 
rail  as  shown  by  the  dotted  lines. 

To  Find  the  Lengths  of  Balusters: — Take  for  example  baluster  1:  1  R  measures  i^",  which 
added  to  2'.2" — the  usual  length  of  balusters  at  X  X  between  the  top  of  step  and  bottom  of  rail — 
makes  the  length  of  baluster  1  between  the  same  points  2'.3^". 

Fig.  3.  Plan  of  Hand-rail,  Quarter-circle  A  E,  Fig.  i,  with  its  Tangents  and  Angles  of 
Inclination  Lettered  Alike. — Connect  M  B,  the  directing  level  line.  Parallel  to  M  B  draw  Q  P; 
parallel  to  E  F  draw  0  N;  through  E  draw  A  U  indefinitely;  on  B  as  centre  with  B  F  as  radius 
describe  the  arc  F  U. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  Both  Joints  : — Pro- 
long BE  to  T;  make  E  T  equal  E  Y;  connect  T  M:  then  the  bevel  at  T  will  contain  the  angle 
required. 

Fig.  4.  Face-mould  from  Plan  Fig.  3,  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints. — Draw  the  line  U  U  indefinitely;  make  V  U,  V  U  equal  V  U  of  FiG.  3.  Draw 
V  B  at  right  angles  to  U  V;  make  V  B  equal  V  B  of  Fig.  3;  connect  B  U  and  B  U;  prolong 
B  U  to  W,  making  U  W  equal  A  N  of  FiG.  2.  The  joints  U  and  W  are  made  at  right  angles 
to  the  tangents.  Make  B  N  and  B  N  each  equal  B  N  of  Fig.  3.  Through  N  and  N  parallel 
to  B  Z  draw  P  R  and  P  R;  make  N  R  and  N  R  each  equal  0  Q  of  Fig.  3;  make  N  P  and 
N  P  each  equal  0  P  of  Fig.  3;  make  V  Z  equal  V  Z  of  Fig.  3.  Through  U  and  U  draw  R  S 
and  R  S,  and  make  U  S  equal  U  R  at  U  and  U;  draw  R  Y  parallel  to  B  U;  from  S  and  R 
draw  lines  to  joint  W  parallel  to  B  W;  through  S  P  B  P  S  of  the  convex  and  R  Z  R  of  the 
concave  trace  the  curved  edges  of  the  face-mould.  The  angle  with  which  to  square  the  wreath- 
piece  at  both  joints  is  taken  by  the  bevel  T  of  Fig.  3.  For  detailed  explanation  of  this  face  mould 
see  Plate  No.  ii.  The  development  of  the  centre  line  of  a  wreath-piece  of  this  kind  is  given  in 
detail  at  Plate  No.  20,  quarter-circle  A  C  of  Fig.  i,  and  AE  of  Fig.  2. 


Plate  No. 47 


Plate  No.48 


c 


PLATE  48. 


Fig-.  I.  Plan  of  the  Bottom  Part  of  a  Flight  of  Winding  Stairs,  Turning  One  Quarter, 
Starting  with  a  Newel. — The  complete  plan  of  this  flight  of  stairs  is  given  at  Plate  No.  7,  Fig.  3. 

Draw  the  centre  line  of  rail,  and  space  the  balusters  as  required  ;  then  to  find  the  length  of  plan 
tangent  A  C,  and  other  measurements,  proceed  to  set  up  tiie  elevation. 

Fig.  2.  Elevation  of  Treads  and  Rises  as  Given  at  Plan ;  also  the  Development  of  the 
Centre  Line  of  the  Wreath-piece. — Place  the  centre  of  baluster  on  each  step  as  at  the  plan,  and 
draw  lines  through  these  centres  parallel  to  the  rise-lines  indefinitely.  Let  the  bottom  line  of 
rail  pass  through  X  X,  the  centres  of  short  balusters  on  the  regular  treads  ;  draw  tlie  centre  line 
of  rail  0  C  parallel  to  X  X  indefinitely  ;  make  B  0  equal  2^' — or  more  at  pleasure — for 
straight  wood  at  the  upper  end  of  the  wreath-piece  ;  make  M  V,  6"  and  V  N  half  the  thickness  of 
rail  ;  through  N  at  right  angles  to  the  chord-line  draw  A  C  :  then  A  C  will  be  the  length  of  the  plan 
tangent  A  C  at  Fig.  i.  And  if  from  C  of  FiG.  i  a  line  is  drawn  touching  the  centre  line  of  rail  at 
M,  it  will  be  the  level  tangent. 

Fig.  3.  Plan  of  the  Centre  Line  of  Rail  and  Tangents  A  C  and  C  M  from  Fig.  i ;  also 
the  Centres  of  Balusters  D,  G,  J  in  Place  as  at  Fig.  i. — Through  A  draw  T  B  at  right  angles  to 
A  C  ;  make  A  B  equal  A  B  of  Fig.  2;  connect  B  C.  T  is  the  centre  of  the  circle,  and  T  M  is  at 
right  angles  to  C  M  as  at  Fig.  i.  Through  the  centres  of  balusters  J,  G,  D,  and  paiallel  to  the 
level  tangent  M  C,  draw  J  K,  V  H  and  S  E;  parallel  to  A  B  draw  E  F,  H  I  and  K  L;  from  A  draw 
A  P  at  right  angles  to  the  level  tangent  CM;  on  C  as  centre  with  C  B  as  radius  describe  the  arc 
B  P  ;  connect  P"  M. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  A : — From 
E  parallel  to  C  B  draw  E  R  indefinitely  ;  prolong  C  A  to  Q  ;  make  A  Q  equal  A  R  ;  connect  Q  S. 
Then  the  bevel  at  Q  contains  the  angle  required. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  M: — Make 
U  V  equal  H  I  ;  connect  V  M.  Then  the  bevel  at  V  contains  the  angle  sought.  Again  at  FiG.  2, 
baluster  D,  make  E  F  equal  E  F  of  Fig.  3;  at  baluster  G,  make  H  I  equal  H  I  of  Fig.  3;  at  baluster 
J,  make  K  L  equal  K  L  of  Fig.  3.  Through  the  points  B  F  I  L  N  trace  the  centre  line  of  the 
wreath-piece;  set  ofT  each  side  of  the  centre  half  the  thickness  of  rail  as  shown  by  the  dotted  lines. 

To  Find  the  Lengths  of  Balusters: — Take  for  example  baluster  D;  D  S  measuring  4^",  which 
must  be  added  to  2'. 2" — the  length  of  short  baluster  at  X  from  the  top  of  step  to  the  bottom  of 
the  rail — making  the  length  of  the  baluster  D  between  the  same  points  2'. 6^".  The  height  of  rail 
at  the  newel  is  calculated  by  adding  M  V,  6"  to  the  length  of  short  baluster  at  X,  2'. 2";  making  the 
height  from  M  to  V  2'. 8". 

Fig.  4.  Parallel  Pattern  for  Wreath-piece  from  Plan  Fig.  3 ;  also  Showing  the  Squaring 
of  the  Wreath-piece  at  the  Joints. — Make  M  B  equal  M  P  of  Fig.  3.  On  B  as  centre  with  B  C 
of  Fig.  3  as  radius  describe  an  arc  at  C;  on  M  as  centre  with  M  C  of  Fig.  3  as  radius  intersect 
the  arc  at  C:  connect  C  M  and  C  B;  prolong  C  B  to  0;  make  B  0  equal  B  0  of  FiG.  2.  Make 
M  W  equal  M  W  of  Fig.  i.  Make  the  joints  0  and  W  at  right  angles  to  the  tangents.  Make 
B  F  I  L  equal  to  B  F  I  L  of  Fig.  3.  Make  F  D,  I  G  and  L  J  equal  ED,  H  G  and  K  J  of  Fig.  3. 
On  the  centres  B  D  G  J  M  with  a  radius  equal  to  half  the  required  width  of  the  pattern 
describe  circles,  and  touching  these  trace  the  edges  of  the  pattern.  The  angle  with  which  to 
square  the  wreath-piece  at  joint  0  is  taken  by  the  bevel  Q  of  Fig.  3,  and  the  angle  for  squaring 
the  wreath-piece  at  joint  W  is  taken  by  the  bevel  at  V  of  Fig.  3.  The  development  of  a  centre 
line  geometrically  the  same  as  tins  of  Fig.  2  is  given  at  Plate  No.  21,  Figs,  i  and  2.  Face-mould 
and  parallel  pattern  as  required  by  this  plan  are  treated  geometrically  in  detail  at  Plate  No.  14. 


PLATE  49 


Fig.  I.  Plan  of  the  Bottom  Portion  of  a  Flight  of  Winding  Stairs,  this  Part  of  the 
Flight  Turning  a  httle  more  than  a  Quarter  and  Starting  from  a  Newel. — Tiiis  case  of 
liaiui-iailing  is  geometrically  ihe  same  as  lluil  given  at  Plate  No.  48.  In  the  last-meniioned 
Plate  the  cylinder  is  10"  in  diameter,  embracing  three  winders;  but  the  plan  here  presented 
has  a  cylinder  20"  in  diameter,  containing  five  winders.  The  object  of  introducing  this  example 
is  to  demonstrate  the  correctness  and  practicability  of  working  the  wreath  aiound  a  large 
cylinder  in  one  piece,  showing,  too,  by  the  development  of  the  centre  line  of  the  wreath  its 
exact  position  and  relation  to  step  and  rise.  The  length  of  plan  tangent  A  B  cannot  be  deter- 
mined until  the  elevation  is  drawn,  if  a  fixed   height  of  rail  at  the  newel  is  required. 

Fig.  2.  Elevation  of  Treads  and  Rises  as  at  Plan;  also  the  Development  of  the 
Centre  Line  of  Wreath-piece. — Let  the  bottom  line  of  rail  pass  through  X  X,  the  centres  of 
slujrt  balusters  on  the  regular  treads;  draw  the  centre  line  of  rail  F  B  parallel  to  X  X  indefi- 
nitely; make  1  E  equal  8",  and  E  G  half  the  thickness  of  rail.  Through  G  draw  B  A  at  right 
angles  to  the  chord-line.  Again,  at  Fig.  i,  continue  the  centre  line  of  rail  to  B,  and  make 
A  B  equal  A  B  of  Fig.  2;  make  A  C  at  right  angles  to  A  B  and  equal  to  A  C  of  Fig.  2;  from 
B  draw  B  Z  tangent  to  the  centre  line  of  rail;  from  H,  the  centre  of  the  cylinder,  draw 
H  0  at  right  angles  to  B  Z;  then  B  0  is  the  level  tangent.  Place  the  balusters  as  required, 
and  number  those  that  come  under  the  wreath.  Through  balusters  2,  3,  4  and  5  draw  S  8, 
F7.  1.6,  5T,  and  from  P,  P  R,  all  parallel  to  OB;  parallel  to  A  C  draw  T  K,  Q  L,  6  N,  7  J 
and  8,  9. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  Z:— 

Make  G  F  equal   7  J;  connect   FO:   then   the   bevel  at   F  contains   the  angle  required. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  A:  — 
From  N  draw  N  M  parallel  to  A  B;  make  AE  equal  M  K;  connect  El:  then  the  bevel  at  E 
contains  the  angle  sought.  From  A  draw  A  D  indefinitely  and  at  right  angles  to  BO;  on  B 
as  centre  with  B  C  as  radius  describe  the  arc  C  D;  connect  D  0.  Again,  at  Fig.  2,  place  the 
centres  of  balusters  on  each  step  as  at  the  plan,  aud  draw  lines  through  these  centres  parallel 
to  the  rise-lines.  At  baluster  2,  make  8,9  equal  8,9  of  Fig.  i;  at  baluster  3,  make  7  J  equal 
7  J  of  Fig.  i;  at  baluster  4,  make  6  N  equal  6  N  of  Fig.  i;  at  baluster  5,  make  T  K  equal  T  K 
3f  Fig.  i;  through  C  K  N  J  9  G  trace  the  centre  line  of  the  wreath-piece;  from  this  centre  set 
off  each  side  half  the  thickness  of  rail  as  shown  by  the  dotted  lines.  Proceed  to  find  the 
lengths  of  balusters  and   the  height  of  rail  at   newel  as  directed  at  Plate  No.  48. 

Fig.  3.  Face-mould  from  Plan  Fig.  i ;  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  the  Joints. — Draw  the  line  CO  equal  to  D  0  of  Fig  i;  on  0  as  centre  with  0  B 
of  Fig.  I  as  radius  describe  an  arc  at  B;  on  C  as  centre  with  C  B  of  Fig.  i  as  radius  intersect 
the  arc  at  B;  connect  0  B  and  B  C;  prolong  B  0  to  Z;  make  0  Z  equal  0  Z  of  Fig.  i;  prolong 
B  C  to  F;  make  C  F  equal  C  F  of  Fig.  2;  make  the  joints  Z  and  F  at  right  angles  to  the  tan- 
gents; make  C  L  N  J  9  equal  the  same  at  Fig.  i;  and  through  each  of  these  divisions  draw  lines 
parallel  to  B  0;  make  0  X  equal  0  X  of  Fig.  i;  make  J  W  V  equal  7  W  Y  of  FiG.  i;  make  N  V  U 
equal  6  V  U  of  Fig.  i;  make  L  P,  L  R  equal  Q  P,  Q  R  of  Fig.  i.  Through  C  draw  P  E;  make  C  F. 
equal  C  P;  make  0  K  equal  0  S;  draw  lines  from  P  and  E  to  the  joint  F  parallel  to  C  B;  draw 
lines  from  K  and  S  to  joint  Z  parallel  to  B  0;  through  E  R  V  W  X  K  of  the  convex  and  P  U  Y  S 
of  the  concave  trace  the  curved  edges  of  the  face-mould.  Joint  Z  of  the  wreath-piece  is  squared 
by  the  angle  contained  in  bevel  F  of  Fig.  r,  and  joint  F  is  squared  by  the  angle  contained  in 
bevel  E  of  Fic.  i. 

Fig.  4.   A  Sketch  of  this  Wreath-piece  as  it  Appears  when  Squared  up. 


Plate  No.  50 


'.       ,  TOP  or  n RST^TEP. 

Scale  ^In.=  1  Ft  ^ 


PLATE  50. 

Hand-rail  over  Steamboat  Stairs,  from  Plan  given  at  Plate  No.  6,  Fig.  9. — Fia.  i,  2  and  3  are 
together  the  plan  of  the  string  with  its  different  curves,  including  the  whole  number  of  treads.  Describe  the 
centre  line  of  rail  and  on  this  centre  line  place  the  balusters  on  each  step  numbered  as  shown.  The 
intention  is  to  take  two  treads  in  the  first  piece  of  rail  from  A  to  C,  and  in  the  second  piece  of  rail  to 
include  six  treads  from  C  to  N;  the  third  piece  of  rail  to  take  the  two  last  treads  with  as  much  more  as 
it  requires  to  bring  this  top  wreath-piece  to  a  level  at  the  required  height.  Draw  the  level  tangent  A  B 
at  right  angles  to  A  J ;  through  C  draw  B  K  at  right  angles  to  J  D;  through  N  at  right  angles  to  N,  43, 
draw  K  0  indefinitely;   the  point  0  must  now  be  established  by  measurement  taken  from  the  elevation. 

Fig.  4.  Elevation  of  Treads  and  Rises  as  given  at  Plan ;  also  the  Development  of  the 
Centre  Line  of  Wreath,  including  the  Whole  Flight. — On  the  line  of  the  third  rise  C  D  fix  D 
distant  from  baluster  3,  so  that  the  bottom  line  of  rail  will  pass  through  3;  draw  D  K  at  right  angles  to 
D  C;  make  D  K  equal  C  K  of  Fig,  2,  draw  K  L  parallel  to  the  rise-lines,  and  equal  to  three  rises;  connect 
L  D,  and  prolong  to  B  indefinitely;  make  C  B  equal  C  B  of  Fig.  i  where  the  length  of  tangent  C  B 
intersects  the  inclined  line  D  B  at  B;  draw  C  A  through  B  at  right  angles  to  CD;  touching  L  draw 
G  N  at  right  angles  to  L  K;  prolong  the  ninth  rise  to  M  and  N;  make  N  G  equal  N  K  of  Fig.  2;  make 
N  M  equal  three  rises;  connect  M  G;  draw  M  0  at  right  angles  to  M  N;  make  40,  41  equal  four  inches; 
make  41,  P  equal  half  the  thickness  of  rail.  Again  at  Figs,  i,  2  and  3,  make  N  M  of  Fig.  2  at  right 
angles  to  K  N,  and  equal  to  three  rises;  then  at  Fig.  3  make  0  P  equal  0  P  of  Fig.  4;  draw  P,  39 
parallel  to  ON;  from  N  parallel  to  K  M  draw  N  P;  from  P  d  raw  P  0  parallel  to  M  N ;  from  0  draw 
the  line  0  T,  touching  the  centre  line  of  rail;  from  the  centre  V  draw  V  S  at  right  angles  to  0  T;  then 
0  S  will  be   the   level   tangent;    parallel   to  0  S  draw   12,  36;  11,  34  and    Z  10,  33;    parallel    to  0  P  draw 

36,  37;  34,  35;  33,  R  and  9,  38. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  from  Fig.  3  over  Joint  N: — 
Make  N,  42  equal  33,  32;  connect  42,  Z:    then  the  bevel  at  42  contains  the  angle  required. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  from  Fig.  3  at  the  Joint  over 
S: — From  R  draw  R  13  parallel  to  N  0;  make  ST  equal  13,  P;  connect  T  U:  then  the  bevel  at  T 
contains  the  angle  sought.  From  N  at  right  angles  to  S  0  draw  N  X  indefinitely;  on  0  as  centre  with 
N  P  as  radius  describe  an  arc  at  X;  connect  X  S.  At  Fig.  2  make  K  L  at  right  angles  to  C  K  and  equal 
to  three  rises;  connect  L  C;  parallel  to  C  L  draw  B  D  of  Fig.  i;  from  C  through  N  draw  C  Q  indefinitely; 
on  K  as  centre  with  K  M  as  radius  describe  the  arc  M  Q;  connect  K,  43;  parallel  to  K  W  draw  8,  27; 
7,  25;  6,  23;  5,  21  and  4,  19;  parallel  to  N  M  draw  27,  28;  25,  26  and  23,  24;  parallel  to  K  L  draw  21,  22; 
19,  20  and  3,  18. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  from  Plan  Fig.  2  at  Both 
Joints:— Draw  28,29  parallel  to  KN;  make  N,  31  equal  29,30;  connect  31,  Y;  then  the  bevel  at  31 
contains  the  angle  required.  At  Fig.  i,  parallel  to  A  B,  draw  1,  16  and  2,  14;  parallel  to  C  D  draw  14,15 
and  16,17;  from  C  draw  C  E  at  right  angles  to  A  B;  on  B  as  centre  with  B  D  as  radius  describe  the 
arc  D  E;  connect  E  A. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  from  Plan  Fig.  i  at  the  Joint 
over  C: — Prolong  line  of  joint  C  D,  and  level  line  A  B,  to  G;  make  C  F  equal  C,  15;  connect  F  G:  then 
the  bevel  at  F  contains  the  angle  required. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  from  Plan  Fig.  i  at  the  Joint 
over  A: — Parallel  to  B  A  draw  C  I;  make  H  I  equal  C  D;  connect  I  A:  then  the  bevel  at  I  contains  the 
angle  sought.  Again  at  Fig.  4:  take  all  the  heights  from  the  plan  tangents  of  Figs,  i,  2  and  3,  and 
place  them  on  the  lines  drawn  through  the  centres  of  like-numbered  balusters,  and  as  shown  by  the 
other  corresponding  numbers  and  letters;  and  through  these  top-numbers  and  letters  trace  the  cei;tre 
line  of  wreath.  The  governing  length  of  baluster  on  this  flight  ought  to  be  2'.4"  at  its  centre,  from 
tr>p  of  step  to  bottom  of  rail.  The  odd  lengths  of  balusters  will  be  found  as  before  explained.  In  case 
the  bottom  line  of  rail  falls  below  the  step  or  floor-line,  at  the  centre  line  of  baluster — as,  for  instance, 
here  at  baluster  11 — then  that  distance  must  be  subtracted  from  the  length  of  the  governing  baluster; 
the  remainder  will  be  the  length  of  baluster  11. 

Fig.  5.  Parallel  Pattern  from  Fig.  3  for  Wreath-piece,  Joining  Level  Rail  at  Top ;  also 
Showing  the  Squaring  of  the  Wreath-piece  at  the  Joints. — Make  S  N  equal  S  X  of  Fig.  3;  on  N  as 
centre  with  N  P  of  Fig.  3  as  radius  describe  an  arc  at  P;  on  S  as  centre  with  S  0  of  Fig.  3  as  radius 
intersect  the  arc  at  P;    connect    S  P  and   P  N;    make    N  38,   R  35,  37  equal    the    same    at    Fig.  3;  make 

37,  12  equal  36,  12  of  Fig.  3;  make  35,  11  equal  34,  11  of  Fig.  3;  make  R  10  equal  33,  10  of  Fig.  3; 
make  the  joints  N  and  S  at  right  angles  to  the  tangents.  Through  N,  38,  1 0,  1  1 ,  1 2  and  S  as  centres, 
with  a  radius  equal  to  half  the  width  of  the  required  pattern,  describe  circles,  and  touching  these  trace 
the  edges  of  the  pattern.  To  square  the  wreath-piece  at  joint  N  take  the  bevel  42,  and  for  squaring 
the  wreath-piece  at  joint  S  take  the  bevel  T. 

Fig.  6.  Parallel  Pattern  for  Wreath-piece  over  Six  Treads ;  also  Showing  the  Squaring-  of 
the  Wreath-piece  at  the  Joints. — Make  W  M  and  W  C  each  equal  W  Q  of  Fig.  2;  draw  W  L  at  right 
angles  to  W  M;  make  W  L  equal  W  K  of  Fig.  2;  connect  L  M  and  L  C;  make  the  joints  C  and  M  at 
right  angles  to  the  tangents;  make  C  18,  20,  22  equal  the  same  at  Fig.  2;  make  L,  24,  26  and  28  equal 
K,  24,  26  and  28  of  Fig.  2;  parallel  to  L  W  draw  28,  8;  26,  7;  24,  6;  22,  5;  20,  4;  make  20,4  equal  19,  4  of 
Fig.  2;  make  22,  5  equal  21,  5  of  Fig.  2;  make  24,  6  equal  23,  6  of  Fig.  2;  make  26,  7  equal  25,  7  of 
Fig.  2;  make  28,  8  equal  27,  8  of  Fig.  2.  Through  C,  18,  4,  5,  6,  7,  8,  M  as  centres  with  a  radius  -equal  to 
one  half  the  width  of  the  required  pattern  describe  circles,  and,  touching  these,  trace  the  curved  edges  of 
the  pattern.      Both  joints  of  this  wreath-piece  are  squared  by  the  bevel  at  31  of  FiG.  2. 

Fig.  7.  Parallel  Pattern  for  Wreath-piece  Joining  the  Newel  at  the  Starting,  and  In- 
cluding the  Two  First  Treads ;  also  Showing  the  Squaring  at  the  Joints  of  the  Wreath-piece.— 
Make  A  D  equal  A  E  of  Fig.  i.  On  D  as  centre  with  D  B  of  Fig.  i  as  radius  describe  an  arc  at  B;  on  A 
as  centre  with  A  B  of  Fig.  i  as  radius  intersect  the  arc  at  B;  connect  B  D  and  B  A;  make  the  joints 
A  and  D  at  right  angles  to  the  tangents;  make  B,  1 7,  1 5  equal  the  same  at  Fig.  i;  make  17,1  and 
15,  2  equal  16,  1  and  14,  2  of  Fig.  i;  through  A,  1,  2,  D  as  centres  with  a  radius  equal  to  one  half 
the  required  width  of  pattern  describe  circles,  and,  touching  these,  trace  the  edges  of  the  pattern.  The 
angle  with  which  to  square  the  wreath-piece  at  joint  D  is  taken  by  the  bevel  F  at  Fig.  t,  and  for  joint 
A  the  angle  is  taken  by  the  bevel  I  of  Fig.  i.  Face-moulds  and  parallel  patterns  are  treated  in  detail  at  the 
following  Plates:  FiG.  5  at  Plate  No.  14;  Fig.  6  at  Plate  No.  15;  Fig.  7  at  Plate  No.  13.  Development  of 
the  centre  lines  of  wreaths  is  given  in  detail  at  the  following  Plates  and  Figures:  Fig.  5  at  Plate  No.  21,  Figs,  i 
a7id  2,  and  of  FiG.  6  at  the  same  Plate,  Figs.  7  and  8;   Fig.  7  at  Plate  No.  20,  Figs.  5  and  6. 


PLATE  51. 

Fig.  I.  Plan  of  Stairs  Showing  how  to  Place  Parallel  Steps  of  a  Uniform  Width  in 
a  Large  Cylinder,  Avoid  Winders,  and  Make  Use  of  the  Room  Afforded  by  Securing  a  Full 
Platform ;  also  an  Evenly-graded  Hand-rail  in  Three  Parts,  Free  from  Abrupt  Top 
Curves  or  Ramps. —  This  pUui  is  given  ixt  Plate  No.  6,  Fig.  8.  Describe  tiie  centre  line  of  rail; 
set  off  and  number  the  balusters  coming  witliin  the  cylinder  as  shown.  Divide  the  cylinder  into 
tliree  equal  parts  by  the  radials  R.  29,  and  R  I;  draw  tangents  to  the  centre  line  of  rail  as  fol- 
lows: At  right  angles  to  R  U  draw  U  16;  through  A  at  right  angles  to  R,  29  draw  16  B;  through 
10  at  right  angles  to  R  1  draw  B  35;  al  tight  angles  to  R  42  draw  43,  35;  at  right  angles  to  A  B 
draw  B  D  indefinitely;  at  right  angles  to  B  35  draw  35,  34  indefinitely;  at  right  angles  to  U  16 
draw  16.  15  indefinitelv.     I'^urthci'  measuicnienls  required  will  be  obtained  from  the  elevation. 

Fig.  2.  Elevation  of  Treads  and  Rises  as  given  at  Plan  Fig.  i ;  also  the  Develop- 
ment of  the  Centre  Line  of  Wreath. — Place  the  centre  of  l^alusler  on  each  step  and  number 
them  as  al  the  plan.  Through  each  of  the  centres  draw  lines  parallel  to  the  rise-lines  indefinitely. 
Let  the  bottom  line  of  rail  at  the  upper  and  lower  ends  pass  through  X  X  and  I  X,  tlie  centres  of 
short  balusters;  parallel  to  X  X  draw  the  centre  line  of  rail  B  34,  parallel  to  1,  X  draw  the  centre 
line  of  rail  A  15;  at  right  angles  to  the  chord-line  draw  U  16;  make  U  16  equal  the  tangent  U  16 
of  Fig.  i;  parallel  to  the  chord-line  draw  16,  15;  parallel  to  U  16  draw  15  F  indefinitely;  make 
43,  34  equal  43,  35  of  Fig.  i;  parallel  to  the  chord-line  draw  34  F;  draw  34,  43  at  right  angles  to 
the  chord-line;  divide  F  34  into  four  equal  parts  and  draw  lines  tlirough  each  division  par- 
allel to  F  15  indefinitely.  Make  42  B  and  U  A  each  three  inches  for  straight  wood  to  be  added 
to  the  upper  end  of  one  and  the  lower  end  of  the  other  wreath-piece,  connecting  with  the 
straight  rail.  Again,  at  Fig.  r,  make  16,  15  equal  16.  15  of  Fig.  2;  connect  1 5,  U ;  make 
43,42  equal  the  same  at  Fig.  2;  connect  42,35.  Set  up  the  following  heights:  35,34;  1 0,  I ; 
B  D  and  A  29 — each  equal  one  of  the  four  equal  heights  at  34  F  of  Fig.  2.  Connect  34,  10; 
I  B,  D  A  and  29,16;  make  42,44  equal  35,34;  draw  44,38  parallel  to  43,35;  make  38,36 
parallel  to  43,42;  diaw  43,37  parallel  and  equal  to  35,  10;  connect  36,37;  parallel  to  36,37 
draw  1  1,30,  1  2,32  and  13.39;  parallel  to  43,42  draw  14,41,  39,40  and  36,38;  parallel  to 
35,  34  draw  32,  33  and  30,31.  Make  16,17  equal  A  29;  draw  17,19  parallel  to  16  U;  draw 
19,  23  parallel  to  16,  15;  make  U  25  parallel  and  equal  to  16  A;  connect  25,  23;  parallel  to 
25.23  draw  4,26,  3,24  and  2,20.  Parallel  to  A  29  draw  5,28  and  26,27;  parallel  to  16,15 
draw  24,  18;  20,  21  and  1,  22.  At  the  middle  piece  of  hand-rail  describe  half  its  width  each 
side  of  the  centre  line.  Through  A  and  10  draw  AO  indefinitely;  at  right  angles  to  A  10  draw 
BP;  parallel  to  B  P  draw  E  F,  Y  T,  8  C  and  7N;  parallel  to  B  D  draw  NM  and  6  L;  parallel 
to   10,  I   draw  X  H,  Z  G   and   C  K;  on    B  as  centre  with    B  I    as   ladius  describe  the  arc  I,  0. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  Both  Joints : — Prolong 
T  Y  to  W;  draw  GJ  parallel  to  10  B;  make  10  S  equal  J  H;  connect  S  W:  then  the  bevel  at 
S  contains  the  angle  required.  Again,  at  Fig.  2,  take  all  the  heights  from  the  plan  tangents  at 
Fig.  I  and  place  them  on  the  lines  drawn  through  the  centres  of  like-numbered  balusters,  and  as 
shown  by  the  other  corresponding  numbers  and  letters;  and  through  the  top  numbers  and  letters 
trace  the  centre  line  of  wreath.    The  odd  lengths  of  balusters  will  be  found  as   before  explained. 

Fig.  3.  Face-mould  for  the  Middle  Piece  of  Hand-rail;  also  Showing  the  Squaring 
of  the  Wreath-piece  at  the  Joints. — Make  PO  and  PA  each  equal  P  0  of  Fig.  i;  make 
P  D  equal  P  B  of  Fig.  i;  connect  A  D  and  0  D;  make  D  G  H  and  D  G  H  equal  B  G  H  of  Fig.  i. 
Parallel  to  P  D  through  G,  H  and  G,  H  draw  E  F  and  Y  T;  make  D  Q  V  equal  B  Q  V  of  Fig.  i; 
make  G  T,  G  Y,  G  T,  G  Y  equal  Z  T,  Z  Y  of  Fig.  i;  make  H  F,  H  E,  H  F,  H  E  equal  X  F,  X  E  of 
Fig  i;  through  A  draw  E  S;  make  A  S  equal  A  E;  through  0  draw  E  B;  make  0  B  equal  0  E. 
Make  the  joints  A  and  0  at  right  angles  to  the  tangents.  Through  S  F  T  Q  T  F  B  of  the 
convex  and  E  Y  P  Y  E  of  the  concave  trace  the  curved  edges  of  the  face-mould.  The  angle  for 
squaring  the  wreath-piece  at  joints  0  and  A  is  taken  by  the  bevel  S  of  Fig.  i. 

Fig.  4.  Plan  of  the  First  Third  of  the  Wreath,  with  the  Tangents  and  Angles  of 
Inclination  from  U  to  A  of  Fig.  I. — Make  U  K  parallel  and  equal  to  16  A;  make  16,  0  equal 
A  29;  make  0  N  parallel  to  U  16;  make  N  M  parallel  to  15,  16;  connect  M  K,  the  directing 
level  line;  parallel  to  M  K  draw  Q  S,  W  L,  16  D,  Z  F  and  J  I;  parallel  to  Q  29  draw  VT  and 
X  Y;  parallel  to  16,  15  draw  2,4  and  6,  3;  at  right  angles  to  K  M  draw  A  B  and  U  H;  on  16 
as  centre  with  16,  29  as  radius  describe  the  arc  29  B;  again,  on  16  as  centre  with  U  1 5  as 
radius  describe  an  arc  at   H;  connect   H  B. 

To  Find  the  Angle  with  which  to  Square  the  Wreath  at  the  Joint  over  A: — Make 
A  E  ecpial  A  T;  connect  E  D:  then  the  bevel  at  E  contains  the  angle  required. 

To  Find  the  Angle  with  which  to  Square  the  Wreath  at  the  Joint  over  U  : — Draw 
F  G  at  right  angles  to  F  U;  make  F  G  equal  2,  5;  connect  G  U:  then  the  bevel  at  G  contains 
the  angle  sought. 

Fig.  5.  Face-mould  over  the  Plan  Fig.  4,  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  the  Joints. — Make  B  C  H  equal  B  C  H  of  Fig.  4;  on  H  as  centre  with  U  1 5  of  Fig.  4 
as  radius  describe  an  arc  at  16;  on  C  as  centre  with  C  16  of  Fig.  4  as  radius  intersect 
the  arc  at  16;  connect  B  16,  C  16  and  16  H;  make  16,  2,  3  equal  1 5,  4,  3  of  Fig.  4;  make  16, 
Y.  T  equal  16,  Y,  T  of  Fig.  4;  parallel  to  C  16,  through  T,  Y,  2,  3,  draw  J  I,  8  Z,  W  L  and  Q  S; 
make  3,  I,  3  J,  2  Z,  2  8  equal  6  I,  6  J,  2  Z  and  2,  8  of  Fig.  4;  make  16,  P,  G  equal  16,  P  9  of 
Fig.  4;  make  Y  L,  Y  W,  T  S  and  T  Q  equal  X  L,  X  W,  V  Q,  V  S  of  Fig.  4.  Through  H  draw  J  E; 
make  H  E  equal  H  J;  through  B  draw  Q  F;  make  B  F  equal  B  Q;  make  H  D  equal  U  A  or 
42,  B  of  Fig.  2.  Make  the  joints  D  and  B  at  right  angles  to  the  tangents.  Draw  lines  from 
E  and  J  to  joint  D  parallel  to  16,  H;  through  E  I  Z  P  L  S  F  of  the  convex  and  J  8  C  W  Q  of 
the  concave  trace  the  curved  edges  of  the  face-mould.  Make  the  slide-line  at  right  angles  to 
the  level  line  C  16.  Joint  B  of  the  wreath-piece  is  squared  by  the  bevel  at  E  of  Fig.  4,  and 
joint  D  is  squared  by  the  bevel  at  G  of  Fig.  4.  A  face-mould  geometrically  the  same  as 
Fig.  3  is  given  in  detail  at  Plate  No.  15,  and  a  face-mould  geometrically  the  same  as  Fig.  5 
is  also  given  in  detail  at  Plate  No.  16.  Development  of  the  centre  line  geometrically  the 
same  as  the  centre  line  of  this  wreath-piece  from  U  to  A  of  Fig.  i  is  given  in  detail  at 
Plate  No.  21,  Figs.  9  and  10;  also  an  example  of  the  development  of  a  centre  line  of  wreath 
from  A  to  10  of  Fig.  i  is  given  geometrically  the  same  in  the  last-mentioned  Plate,  Figs  7 
and  « 


Plate  No.  51  . 


Plate  No.  52 


Scale  f  i  n.  =  1  ft. 


PLATE  52. 


Fig.  I.  Plan  of  a  Platform  and  Double-landing  Steamship  Staircase  with  Newels  at 
the  Starting,  at  the  Angles  of  the  Platform  and  at  the  Landings. — The  posts  at  the  starting 
are  intended  to  run  above  the  upper  deck  a  sufficient  height  to  receive  the  level  hand-rail  and 
balustrade  of  that  deck  as  shown  by  the  dotted  lines.  All  the  newel-posts  are  to  be  finished 
above  the  hand-rails  with  moulded  caps,  the  landing-post  also  finished  with  moulded  drops 
below  the  strings.  The  platform-posts  are  to  rest  on  the  lower  deck.  A  plan  of  a  staircase 
similar  to  this  with  a  continued  hand-rail  is  given  at  Plate  No.  6,  Fig.  9,  the  hand-rail  of 
which  is  treated  at  Plate  No.  50. 

Fig.  2.  Elevation  of  Treads  and  Rises  between  the  Starting  Newel  and  the  Plat- 
form Newel. — The  treads  in  the  curve  from  A  to  F  on  the  plan  must  be  measured  on 
the  centre  line  of  rail  in  the  manner  before  directed, — taking  each  tread  in  two  parts.  Place 
the  centres  of  the  three  balusters  in  the  curve  as  numbered  in  position  on  each  tread,  and 
through  these  draw  lines  parallel  to  the  rise-lines  indefinitely.  Make  A  E  equal  A  D  of  Fig.  i 
through  E  draw  E  D  parallel  to  the  rise-lines;  make  D  F  equal  D  F  of  Fig.  i;  continue  the 
tread-line  M  to  F  and  Z,  from  X,  the  centre  of  baluster,  with  a  radius  equal  to  half  the 
thickness  of  rail  describe  an  arc  at  U;  touching  U  draw  a  line  to  F;  at  E  draw  the  line  EA 
at  right  angles  to  E  D;  through  X  parallel  to  U  F  draw  the  bottom  line  of  rail  X  C;  make  B  N 
equal  3"  for  straight  wood  to  be  left  on  the  upper  end  of  the  wreath-piece.  At  right  angles  to 
F  E  draw  F  W;  make  F  W  equal  half  the  thickness  of  rail;  draw  W  Y  parallel  to  E  F. 

Fig.  3.  Plan  of  Rail  with  Centre  Line  and  Tangents  taken  from  A  to  F  of  Fig.  i. — 
Make  A  B  at  right  angles  to  A  D  and  equal  A  B  of  Fig.  2.  Connect  B  D;  at  right  angles  to 
D  F  draw  D  E  equal  to  D  E  of  Fig.  2;  connect  E  F;  connect  D  K;  through  A  draw  F  C  indefi- 
nitely; on  D  as  centre  with  D  B  as  radius  describe  the  arc  B  C.  The  numbers  1,  2  and  3  desig- 
nate the  centres  of  the  first  three  balusters  as  placed  on  the  treads  at  Fig.  i.  Parallel  to 
G  D  draw   P  Q,  3,  0  and  2  Z;  parallel  to  D  E  draw  Z  T  and  1  R. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  Both  Joints  Pro- 
long D  F  to  J  indefinitely;  make  FJ  equal  D  H;  connect  J  K;  then  the  bevel  at  J  contains 
the  angle  sought.  Again,  at  Fig.  2,  take  the  heights  0  V,  Z  T  and  1  R  from  Fig.  3  and  place 
them  at  the  like-numbered  balusters  as  shown  ;  then  through  M  R  T  V  B  trace  the  centre-line 
of  wreath-piece;  below  this  centre  line  set  off  half  the  thickness  of  rail  for  the  bottom  line 
of  wreath.  Find  the  odd  lengths  of  balusters  as  before  explained,  first  fixing  the  length  of 
baluster  at  X  to  suit,  which  should  not  be  less  than  2'. 4"  from  top  of  step  to  the  bottom  of 
rail  at  the  centre  of  baluster. 

Fig.  4.  Face-mould  from  Plan  Fig.  3;  also  Showing  the  Squaring  of  the  Wreath  at 
the  Joints. — Make  G  C.  G  C  each  equal  G  C  of  Fig.  3.  Draw  G  D  at  right  angles  to  G  C; 
make  G  D  equal  G  D  of  Fig.  3;  connect  D  C,  D  C,  and  prolong  each  indefinitely;  make  C  Y 
equal  W  Y  of  Fig.  2;  make  C  N  equal  B  N  of  Fig.  2;  make  the  joints  N  and  Y  at  riglit  angles 
to  the  tangents;  make  D  U,  D  U  each  equal  D  U  of  Fig.  3.  Parallel  to  G  D  draw  U  P,  U  P; 
make  U  P,  U  P  each  equal  Q  P  of  Fig.  3;  make  D  X  X  equal  D  X  X  of  Fig.  3;  through  C  and 
C  draw  P  L  and  P  L;  make  C  L  and  C  L  equal  C  P;  draw  lines  from  P  and  L  to  the  joints 
parallel  to  the  tangents;  through  L  X  L  of  the  convex  and  P  X  P  of  the  concave  trace  the 
curved  edges  of  the  face-mould.  The  dotted  lines  show  the  extra  width  of  wood  required — 
greater  than  the  width  of  face-mould — to  get  out  the  wreath-piece  with  this  proportioned  form 
of  hand-rail.  This  wreath-piece  is  squared  at  the  joints  Y  and  N  by  the  angle  at  bevel  J  of 
Fig.  3.  An  elementary  study  of  a  face-mould  geometrically  the  same  as  this  is  given  at  Plate 
No.  15;  also  a  like  study  of  the  development  of  a  centre  line  of  wreath  geometrically  the 
same  as  at  Fig.  2  is  given  at  Plate  No.  21,  Figs.  7  and  8. 


PLATE  53. 

Hand-rail  for  Circular  Staircase  from  Plan  given  at  Plate  No.  7,  Fig.  ic— Figs,  i,  2 
and  3  are  together  the  plan  of  the  string  with  its  curves,  including  the  whole  number  of 
treads.  Describe  the  centre  line  of  rail  |''  greater  radius  than  the  front-string.  The  hand-rail 
of  this  flight  is  divided  into  five  parts:  Fig.  1  from  the  newel-post  A  to  D  embraces  three 
treads,  and  three  more  divisions  of  the  hand-rail  will  each  include  five  treads;  the  fifth  piece 
of  lail  will  take  the  last  tread,  and  as  much  more  of  the  curve  as  it  requires  to  bring  this 
top  wreath-piece  to  a  level  at  its  proper  height.  Draw  the  tangents  to  the  centre  line  of 
rail  as  follows-  At  right  angles  to  the  radial  YE,  touching  D,  draw  B  1;  at  right  angles  to 
the  radial  Y  C,  touching  C,  draw  1  F;  at  right  angles  to  the  radial  Y  L,  touching  L,  draw  4  F; 
at  right  angles  to  Y  X,  touching  X,  draw  U  4;  at  right  angles  to  D  1  draw  1,2;  make  1,  2 
equal  two  rises  and  a  half,  the  rise  being  "jY;  connect  2  D;  make  C  3  and  F  E  each  at  right 
angles  to  1  F,  and  each  equal  to  two  and  a  half  rises;  connect  3,  1,  also  E  C;  at  right  angles 
to  F  4  draw  4,5;  make  4,5  and  LJ  each  equal  two  and  a  half  rises;  connect  5  L,  also  J  F; 
make  X  6  equal  two  and  a  half  rises;  connect  6,  4.  At  Fig.  2,  through  L  draw  C  K  indefinitely; 
on  F  as  centre  with  F  J  as  radius  describe  the  arc  J  K;  from  M  parallel  to  Y  F  draw  M  N, 
Q  R  and   H  F;  parallel  to  L  J  draw  0  T  and  P  S. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  Both  Joints:— 
C  and  L,  Fio.  2.  Parallel  u>  F  Y  draw  A  G ;  at  right  angles  to  Y  C  draw  A  B;  make  A  B  equal  G  D;  con- 
nect B  C:  then  tlie  bevel  at  B  contains  the  angle  sought.  At  Fig.  i  the  curve  of  string  which  includes 
the  two  first  treads  has  a  radius  I  Z  equal  to  one  foot;  also,  the  limit  of  tangents  D  B  and  B  A 
cannot  be  determined  until  a  portion  of  the  elevation  is  set  up;  neither  can  the  tangents  X  U 
and  U  B  of  Fig.  3  be  fixed,  for  the  same  reason. 

Fig.  4.  A  Portion  of  an  Elevation  of  Treads  and  Rises,  including  the  Three  First 
Treads,  the  Top  Tread,  and  Landing. — Let  the  bottom  line  of  rail  pass  through  XXX,  the 
centres  of  balusters;  make  W  T  equal  8",  and  T  Z  half  the  thickness  of  rail;  draw  Z  D  parallel 
to  line  of  tread;  prolong  the  fourth  line  of  rise  to  C  and  D;  draw  C  U  at  right  angles  to  C  D 
indefinitely;  make  S  E  equal  4",  and  E  V  half  the  thickness  of  rail.  Again  at  Fig.  i,  make  D  C 
equal  D  C  of  FiG.  4;  parallel  to  2  D  draw  C  B;  from  B  draw  the  tangent  B  A,  touching  the 
centre  line  of  rail;  from  I  at  right  angles  to  B  A  draw  I  A;  from  D  at  right  angles  to  B  A 
draw  D  H  indefinitely;  on  B  as  centre  with  B  C  as  radius  describe  the  arc  C  H;  connect  H  A; 
parallel  to  B  A  draw   P  Q,  R  X  and  V  W;    parallel  to   D  E  draw  0  F,  S  M  and  U  N. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  D: — 
Fig.  I.  Parallel  to  A  B  prolong  RX  to  E;  from  S  draw  SL  parallel  to  B  C;  make  DG  equal  D  L; 
connect  G  E:  then  the  bevel  at  G  contains  the  angle  sought. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  A:— Fig.  i. 
Let  D  J  be  parallel  to  B  A;  make  K  J  equal  D  C;  connect  J  A:  then  the  bevel  at  J  will  contain 
the  angle  required.  At  Fio.  3,  make  U  V  equal  U  V  of  Fig.  4;  draw  V  W  parallel  to  U  X;  from 
X  parallel  to  6,4  draw  XV;  from  V  at  right  angles  to  X  U  draw  V  U;  from  U  draw  U  C, 
touching  the  centre  line;  from  Y  at  right  angles  to  U  C  draw  Y  B;  from  X  at  right  angles 
to  U  B  draw  X  H  indefinitely;  with  X  V  as  radius  on  U  as  centre  describe  an  arc  at  H; 
parallel  to  B  U  draw  A  G,  E  K  and  X  D. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  X:- 
Make  X  F  equal  Z  0;  connect   F  G:    then  the  bevel  at  F  contains  the  angle  sought. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  B: — 
Fio.  3.     Make  B  C  equal  U  V;  connect  C  D:  then  the  bevel  at  C  contains  the  angle  required. 

Fig.  5.  Face-mould  from  Plan  Fig.  i ;  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  the  Joints. — Make  C  A  equal  A  H  of  Fig.  i.  On  C  as  centre  with  C  B  of  Fig.  i  as 
radius  describe  an  arc  at  B;  on  A  as  centre  with  A  B  of  Fig.  i  as  radius  intersect  the  arc 
at  B;  connect  C  B  and  B  A;  make  C  F  M  N  equal  the  same  at  Fig.  i;  parallel  to  A  B  through 
F  M  and  N  draw  V  W,  R  S  and  Q  P;  make  F  Q  and  F  P  equal  0  Q  and  0  P  of  Fig.  i;  make 
M  R  and  M  S  equal  S  R  and  S  X  of  Fig.  i;  make  N  W  equal  U  W  of  Fig.  i;  through  C  draw 
P  T;  make  C  T  equal  C  P;  make  the  joints  A  and  C  at  right  angles  to  the  tangents;  make 
A  E  equal  A  V;  through  P  S  W  B  E  of  the  convex  and  T  Q  R  V  of  the  concave  trace  the  curved 
edges  of  the  face-mould.  This  wreath-piece  is  squared  at  the  joint  C  by  the  angle  at  bevel 
G  of  Fig.  i,  and  at  joint  A  is  squared  by  the  angle  at  bevel  J   of  Fig.  i. 

Fig.  6.  Face-mould  from  Plan  Fig.  2;  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  the  Joints. — Make  H  K,  H  K  each  equal  H  K  of  Fig.  2;  make  H  F  at  right  angles  to  H  K, 
and  equal  to  H  F  of  Fig.  2;  connect  F  K  and  F  K;  make  EST  and  EST  equal  the  same  at 
Fig.  2;  parallel  to  H  F  through  ST  and  ST  draw  M  N,  Q  R  and  M  N,  Q  R;  through  K  and  K 
draw  M  Z  and  M  Z;  make  K  Z,  K  Z  each  equal  M  K;  make  F  0  0  equal  F  X  X  of  Fig.  2;  make 
S  R,  S  Q,  T  N,  T  M  each  side  of  the  centre  0  0  equal  P  R,  P  Q,  0  N  and  0  M  of  Fig.  2;  make 
the  joints  K,  K  at  right  angles  to  the  tangents;  through  ZNRORNZ  of  the  convex  and 
M  Q  0  Q  M  of  the  concave  trace  the  curved  edges  of  the  face-mould.  This  wreath-piece  is 
squared  at  both  joints  by  the  angle  at  bevel   B  of  Fig.  2. 

Fig,  7.  Face-mould  from  Plan  Fig.  3  ;  also  Showing  the  Squaring  of  the  Wreath- 
piece  at  Both  Joints. — Let  B  X  equal  B  H  of  Fig.  3.  On  B  as  centre  with  B  U  of  Fig.  3  as 
radius  describe  an  arc  at  U ;  on  X  as  centre  with  X  V  of  Fig.  3  as  radius  intersect  the  arc 
at  U;  connect  X  U  and  U  B;  make  the  joints  B  and  X  at  right  angles  to  the  tangents;  make 
XOL  equal  X  0  L  of  Fig.  3;  parallel  to  B  U  and  through  X,  0,  L  draw  X  J,  T  G  and  AM; 
make  U  N  and  L  M  equal  U  P  and  Z  S  of  Fig.  3;  make  X  J,  0  T,  0  G  equal  X  J,  N  K,  N  E  of 
Fig.  3;  make  B  C  equal  B  A;  through  X  draw  G  F;  make  X  F  equal  X  G;  through  G  M  N  C  of 
the  convex  and  A  T  J  F  of  the  concave  trace  the  curved  edges  of  the  face-mould.  The  angle 
with  which  to  square  the  wreath-piece  at  joint  X  is  taken  by  the  bevel  F  at  Fig.  3,  and  for 
joint  B  the  angle  is  taken  by  the  bevel  at  C.  An  elementary  study  of  a  face-mould  geometrically 
the  same  as  Figs.  5  and  7  is  given  at  Plate  No.  13,  and  of  face-mould  Fig.  6  at  Plate  No. 
15.  A  like  study  of  the  development  of  a  centre  line  of  wreath-piece  geometrically  the  same 
as  required  for  Figs.  5  and  7  is  given  at  Plate  No.  20,  Figs.  5  and  6;  also  the  development 
of  a  centre  line  of  wreath-piece  geometrically  the  same  as  required  for  Fig.  6  is  given  at 
Plate  No.  21,  Figs.  7  and  S. 


Plate  No. 53 


LATt  No.  54 


SCAUC      I      1  N.  =  1   F  T. 


PLATE  54. 


Fig.  I.  P)an  of  Starting  the  Circular  Staircase  given  at  Plate  No.  53,  with  a  Scroll 
Step  and  Hand-rail  instead  of  a  Newel. — The  first  three  steps  in  this  plan  are  all  included  in 
the  curve  of  the  scroll,  but  the  bottom  step  is  properly  the  scroll  step  The  radius  Y  D  is  the 
same  as  that  of  the  plan  of  circular  string,  Plate  No.  53.  D  2,  1  is  equal  to  D  2,  1  at  the  plan, 
Plate  No.  53.  Touching  D,  the  tangents  1  A  are  at  right  angles  to  YD;  at  right  angles  to  D  A, 
touching  the  centre  line  of  rail  at  F,  draw  A  F;  make  the  joint  F  at  right  angles  to  F  A;  from 
A  parallel  to  D  2  draw  A  E;  parallel  to  V  E  draw  U  H,  I  C  and  G  B. 

To  Ascertain  the  Height  of  the  Scrolled  Hand-rail,  as  Regulated  by  the  Tangent  D  A 
and  the  Angle  of  Inclination  D  E  A: — Set  up  an  elevation  of  the  first  three  treads,  including  the 
fourth  rise  as  at  Fig.  2.  Let  X  be  the  centre  of  baluster,  and  let  the  bottom  line  of  rail  pass 
through  X;  also  let  D  E  A  equal  D  E  A  of  Fig.  i.  Make  A  B  half  the  thickness  of  rail;  then 
B  C,  which  is  4V',  added  to  whatever  height  of  baluster  is  given  at  X,  will  be  the  total  height 
of  scroll  between  the  top  of  the  first  step  C  and  the  bottom  of  tlie  scroll  B  when  the  rail 
is  set  up.  The  scroll  looks  best  when  kept  at  a  height  between  C  and  B  not  exceeding  2'.6". 
In  shaping  the  top  and  bottom  of  the  scroll  it  is  desirable  not  to  finish  to  a  level  at  the  joint 
F,  but  to  continue  the  easing  an  inch  or  two  lower  down,  coming  to  a  level  with  its  ease- 
ment at  about  the  eye  of  the  scroll.  The  scroll  may  also  be  brought  lower  by  increasing  the 
length  of  tangent  D  A  of  Fig.  i,  forming  an  acute  angle  with  the  plan  tangents;  or  it  may 
be  fixed  at  a  greater  height  by  lessening  the  length  of  tangent  D  A,  and  forming  an  obtuse 
angle  with  the  plan  tangents 

Fig.  3.  Face-mould  from  Plan  of  Scroll  Fig.  i;  also  Showing  the  Squaring  of  the 
Wreath-piece  at  the  Joints. — Let  E  A  equal  E  A  of  Fig.  i;  make  A  F  at  right  angles  to  A  E 
and  equal  to  A  F  of  Fig.  i;  make  E  H  C  B  equal  the  same  at  Fig.  i;  through  E  H  C  B  parallel  to 
F  A  draw  V  V,  U  M,  I  L  and  G  B;  make  E  V,  E  V,  H  U,  H  M,  C  I,  C  L,  B  K  and  A  J  equal  D  V,  T  U, 
T  M,  0  !,  0  L,  S  K  and  A  J  of  Fig.  i.  Through  F  draw  G  P  parallel  to  A  E;  make  F  P  equal 
F  G;  through  P  J  K  L  M  V  of  the  convex  and  V  U  I  G  of  the  concave  trace  the  curved  edges 
of  the  face-mould.  This  wreath-piece  is  squared  at  the  joint  F  by  the  angle  at  bevel  E  of 
Fig.  I.    At  joint  E  the  sides  of  the  wreath  are  at  right  angles  to  the  plane  of  the  plank. 

Fig.  4.  To  Draw  a  Scroll  suitable  for  this  Hand-rail  and  Staircase. — Describe  a  circle 
of  a  diameter  about  sufficient  to  enclose  the  spiral  line  to  be  developed — its  exact  diameter 
is  unimportant.  Divide  the  circumference  of  the  circle  into  sixteen  parts;  make  the  diameter 
of  the  eye  of  the  scroll  S  J  equal  the  width  of  the  hand-rail.  The  spiral  curve  is  found  by 
points  on  these  sixteen  radii,  beginning  at  J  by  drawing  a  line  at  right  angles  to  the  radial 
A  J,  then  at  right  angles  to  the  next  radial  on  the  left,  and  so  on  as  shown  by  the  position 
of  the  little  trying-squares,  the  external  angles  of  which  designate  points  on  which  as  centres 
with  half  the  width  of  rail  as  radius  describe  arcs  of  circles.  Touch  ing  these  arcs  trace 
the  curved  edges  of  the  scroll;  but  at  the  point  0  where  the  arc  touches  the  eye  of  the 
scroll  measure  on  the  radii  from  the  external  angles  of  the  squares  to  the  circle  forming  the 
eye,  and  set  off  these  distances  outward  as  at  0  0,  S  S,  etc.,  to  J,  tracing  the  remainder  of 
the  convex  curve  from  0  to  J  through  the  points  thus  found.  The  spiral  drawn  in  this  way 
may  be  cut  off  at  any  point  where  a  sufficient  revolution  is  made.  In  this  case  it  is  cut  off 
at   D  and  connected  with  the  plan  at   D.   Fig.  i. 

Fig.  5.    Construction  of  Block  for  Scroll  Step  and  Riser. 

Fig.  6.   Scroll  Step  as  Completed. 


PLATE  55. 

Hand-rail   over  Elliptic  Staircase  from   the    Plan  given  at   Plate    No.  7,  Fig.   ii.— The  best 

division  of  hand-rail  lor  this  plan  is  to  begin  at  the  centre,  Fu;.  i,  taking  in  this  first  piece  a  portion  of  rail  including 
three  treads  each  side  of  the  minor  axis;  then  FiG.  3,  covering  four  more  treads,  FiG.  5,  also  taking  four  treads;  and 
Fig.  8,  with  the  bottom  tread  and  as  much  more  of  the  curve  as  may  be  required  to  join  tiie  level  rail  at  its  usual 
iieight.  After  deciding  on  the  proper  number  of  pieces  in  which  to  divide  the  plan  of  hand-rail,  draw  the  tangents  for 
the  whole  as  follows:  Make  B  V  tangent  to  the  centre  of  rail  at  V,  draw  B  N  tangent  to  the  centre  line  of  rail  at  Y, 
draw  NA  tangent  to  the  centre  line  of  rail  at  W  ;  draw  AL  tangent  to  the  centre  line  of  rail  at  C  ;  the  level  tangent 
L  F  will  be  fixed  further  on. 

Fig.  I,  Plan  of  Wreath-piece  including  Six  Treads.— Draw  B  X  at  right  angles  to  B  Y  and  equal  to  three 
rises;  connect  X  Y,  at  right  angles  to  B  V  draw  V  A  ccjual  to  three  rises,  join  A  B,  prolong  A  V  to  F  indefinitely,  from 
Y  through  V  draw  Y  D;  on  B  as  centre  with  B  A  as  radius  describe  the  arc  A  D;  parallel  to  B  N  draw  M  K,  S  R  and  W  T: 
parallel  to  V  A  draw  U  5,  Q  G  and  L  J. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  Both  Joints :— Prolong  RS  to  F,  par- 
allel to  B  V  draw  G  H;  make  V  E  (  (iiial  H  C;  join  E  F.  then  the  bevel  at  E  coiuains  the  angle  sought. 

Fig.  2.    Face-mould  from  Fig.  i ;  also  Showing  the  Squaring  of  the  Wreath-piece  at  the  Joints. 

—  Let  N  A  and  N  A  each  equal  D  N  of  FiG.  i  ,  make  N  B  at  right  angles  to  N  A  and  cqiml  to  N  B  of  Fig.  i  ;  join  B  A 
and  B  A;  make  B  J  G  5  equal  the  same  at  FlG.  i.  Through  J  G  5  draw  K  M,  R  S  and  T  W  par.illel  to  N  B  ;  make  B  P, 
J  K  and  J  M  equal  B  P,  L  K  and  L  M  of  FiG.  i  ;  make  G  R,  G  S,  5  T  and  5  W  equal  Q  R,  Q  S,  U  T  and  U  W  of  FiG.  i. 
Apply  the  same  measurements  the  other  side  of  the  centre  B  P,  and  through  all  tliese  points  trace  the  curved  edges 
of  the  face-mould.  Make  the  joints  A  A  at  right  angles  to  the  tangents.  This  wreath-piece  is  squared  at  both  joints 
by  the  angle  at  bevel  E  f)f  Fig.  i. 

Fig.  3.    Plan  of  Wreath-piece  including  Four  Treads. — Draw  Y  R  at  right  angles  to  Y  N  and  equal  to  four 

rises.  Parallel  to  Y  X  draw  N  I  ;  at  right  angles  to  A  N  draw  W  5  and  N,  14  ;  make  N.  14  equal  1  R  ;  join  14,  W  ;  draw  W  J 
parallel  ami  equal  to  NY;  make  1  S  equal  14,  N  ;  parallel  to  Y  N  draw  S  G  ;  make  G  8  parallel  to  1  Y';  join  8  J  parallel  to 
8  J  draw  T  M.  2,  9  N  K.  7,  22  and  10.  11  ,  parallel  to  Y  I  draw  O  F  and  D  3  :  parallel  to  N,  14  draw  U  P  and  C  I  ;  at  right 
angles  to  J  8  di  aw  Y  Z;  on  N  as  centre  with  N  I  as  radius  describe  the  arc  1  Z  ;  at  right  angles  to  J  8  draw  W  B  ,  on  N  as 
centre  with  W  14  as  radius  describe  an  arc  at  B  ;  join  B  Z. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  Y;— Draw  3  A  parallel 

to  NY,  make  Y  E  equal  A  4;  join  E  2  ;  then  the  bevel  at  E  contains  the  angle  sought. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  W  :— Parallel  to  J  8 

prolong  7,  22  to  5 ,  make  W  X  equal  U  Q;  join  X  5;  then  the  bevel  at  X  contains  the  angle  required. 

Fig.  4.    Face-mould  from  Plan  Fig.  3  ;  also  Showing  the  Squaring  of  the  Wreath-piece  at  the 

Joints.  —  Make  Z  L  B  equal  Z  L  B  of  FiG.  3 ;  on  B  as  centre  with  W,  14  of  Fk;.  3  as  radius  describe  an  arc  at  N ;  on  L  as 
centre  wntli  L  N  of  Fk;.  3  as  radius  intersect  the  arc  at  N  ;  join  B  N,  N  Z  and  L  N;  make  B  I  P  equal  W  I  P  ol  FiG.  3; 
make  N  3  F  equal  N  3  F  of  FiG.  3;  through  I  P.  3  F  parallel  to  L  N  draw  11.  10,  7,  22,  H  9  and  T  M  ;  make  I  10,  I  11,  P  22, 
P  7,  N  12.  N  K  equal  C  10,  C  11,  U  22,  U  7,  N  12,  and  N  K  of  Fig.  3;  make  3  9,  3  H.  F  M,  F  T  equal  D  9,  D  H,  O  iVi  and  O  T  of 
Fig.  3.  Through  Z  dr.iw  T  A;  make  Z  A  equal  Z  T,  make  the  jcjints  B  and  Z  at  right  angles  to  the  tangents;  through 
A  M.  9,  12.  22  and  10  of  the  convex  and  T  H  K  7,  11  o(  the  concave  trace  the  curved  edges  of  the  face-mould.  The  angle 
with  which  to  square  the  wreath-piece  at  joint  Z  is  taken  by  bevel  E,  Fig.  3,  and  for  squaring  joint  B  the  bevel  X  of  Fig.  3. 

Fig.  5-  —  Make  W  T  equal  four  rises.  From  A  make  A  6  parallel  to  W  14  ;  make  A  B  perpendicular  to  A  C  and  equal 
to  6  T ,  join  B  C.  This  position  of  the  plan  with  its  tangents  and  angles  oi  inclination  is  reino\  ed  and  completed  at 
Fig.  6. 

Fig.  6.  Plan  of  Wreath-piece  over  Four  Treads,  taken  from  Fig.  5.— Make  c  N  parallel  and  equal  to 

A  W  ;  make  6  J  etjual  to  A  B  ;  make  J  H  parallel  to  W  A  and  H  R  parallel  to  W  6  ,  join  R  N  and  prolong  to  IVI  indefinitely; 
prolong  6  W  to  M  ;  parallel  to  R  N  draw  Z  4,  K  3  and  A  X  ;  parallel  to  W  6  draw  2  F;  parallel  to  A  B  draw  O  Q  ;  at  rit;iit 
angles  to  N  R  draw  W  E  and  C  D  indefinitely;  on  A  as  centre  with  A  6  as  radius  describe  the  arc  6  E;  again,  on  A  as 
centre  with  B  0  as  radius  describe  an  arc  at  D;  join  D  E. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  w :— Make  W  L  equal 

J  G  ;  join  L  IV!  :  then  the  bevel  at  L  contains  the  angle  sought. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  0 :— Draw  CX 

at  right  angles  to  C  A  ;  draw  X  Y  at  right  angles  to  X  C  and  equal  to  A  V  ;  join  Y  C:  then  the  bevel  at  Y  contains  the 
angle  required. 

Fig.  7.  Face-mould  from  Plan  Fig.  6,  Showing  also  the  Squaring  of  the  Wreath-piece  at  the 
Joints. — Make  DUE  equal  D  U  E  of  FiG.  6;  on  D  as  centre  with  C  B  of  Fig.  6  as  radius  describe  an  arc  at  A ;  on  U  as 
centre  with  U  A  of  FlG.  6  as  radius  intersect  the  arc  at  A;  join  A  E.  A  D  and  U  A  ;  make  DQ  equal  C  Q  of  FiG.  6; 
make  AHF  etjual  A  H  F  of  FiG.  6;  parallel  to  U  A  through  Q  draw  Z4;  parallel  to  U  A  through  H  and  F  draw  PS 
and  K  3  :  make  Q  Z.  Q  4  and  A  T  equal  O  4,  O  Z  and  A  T  of  Fig.  6  ;  make  H  S,  H  P,  F  3  and  F  K  equal  R  S,  R  P,  2,  3 
and  2K  of  FiG.  6;  through  D  draw  ZB;  make  DB  equal  DZ;  through  E  draw  KG;  make  EC  equal  EK;  through 
B  4  A  S  3  C  of  the  convex  and  Z  T  P  K  of  the  concave  trace  the  curved  edges  of  the  face-mould.  The  angle  with 
which  to  square  the  wreath-piece  at  joint  D  is  taken  by  the  bevel  at  Y ;  and  for  joint  E  by  the  bevel  at  L  of  FiG.  6. 

Fig.  8.  Plan  of  Wreath-piece,  including  the  First  Tread. — To  find  the  height  c  O  set  up  an  eleva- 
tion of  the  bottom  step  and  two  rises  as  at  FiG.  9,  let  X  be  the  centre  of  baluster  and  XO  half  the  thickness  oi 
rail.  Make  HZ  equal  four  inches  and  ZL  half  the  thickness  of  rail;  draw  LC  parallel  to  the  floor-line;  the  anL:le 
at  O  must  equal  the  angle  B  of  FiG.  5.  Again  at  FiG.  8  make  CO  equal  CO  of  FiG.  9;  from  O  parallel  to  C  B 
draw  OL;  from  L  draw  LF  tangent  to  the  centre  line  of  rail  at  F;  make  F  H  at  right  angles  to  F  L  ;  parallel  to 
FL  draw  C  G,  U  T  and  JR;  parallel  to  CO  draw  SP  and  RQ;  at  right  angles  to  FL  draw  C  D ,  on  L  as  centre 
with   LO  as  radius  draw  the  arc  OD;  join   D  F. 

To   Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  C;— Make  CK 

equal  CP;  join   KE:  then  the  bevel  at  K  contains  the  angle  sought. 

To  Find  the  Angle  with  which  to  Square  the  Wreath-piece  at  the  Joint  over  F :— Make  H  G 

equal  CO;  join  G  F:  then  the  bevel  at  G  contains  the  angle  required. 

Fig.  10.    Face-mould  from  Fig.  8,  Showing  also  the  Squaring  of  the  Wreath-piece  at  the 

Joints. — Let  F  D  equal  F  D  of  Fig.  8  ;  on  F  as  centre  with  F  L  of  Fig.  8  as  radius  describe  an  arc  at  L;  on  D  as  centre 
with  O  L  of  Fig.  8  as  radius  intersect  the  arc  at  L;  join  F  L  and  L  D;  make  L  X  X  equal  L  Q  P  of  Fig.  8;  through  X  X 
and  D  parallel  to  F  L  draw  X  J,  T  U  and  D  N  ;  make  L  M.  X  T,  X  U  and  D  N  equal  L  M,  S  T,  S  U  and  C  N  of  Fig.  8. 
Make  the  joints  at  right  angles  to  the  tangents;  make  FS  equal  FJ;  through  D  draw  UP;  make  DP  equal  DU; 
through  S  'M  Q  U  of  the  convex  and  J  T  N  P  of  the  concave  trace  the  curved  edges  of  the  face-mould.  This  f:ice- 
mould  will  not  answer  for  the  top  of  the  flight,  because  at  the  top,  although  including  but  one  tread  as  at  the  bottom 
and  setting  up  the  usual  height  from  the  floor  for  the  level  rail,  yet  the  total  height  is  greater.  This  will  be  under 
stood  by  examining  FlG.  11 — which  is  set  up  for  the  top — and  comparing  it  with  FiG.  9. 


Plate  No. 56 


F  I  G  3 


PLATE  50. 

Figs.  I  and  2.  Wreath-piece  from  a  Face  mould,  with  Tangents  at  Right  Angles,  the  Position 
of  one  of  which  Tangents  B  L  is  Inclined,  while  that  of  L  D  is  Horizontal. — Tlie  sliding  of  the  face- 
mould  along  ihe  joint  C  D  and  at  F,  the  other  side  of  the  stuff,  to  plumb  the  sides  of  the  wreath-piece, 
is  sliovvn  by  the  dotted  lines.  The  sides  G  H,  J  K  of  the  wreath-piece  at  the  centre  butt-joint  are  not 
straight  lines.  On  the  concave  side  of  the  wreath-piece  G  H  is  a  concave  curve,  and  J  K  of  the  convex  a 
convex  curve  on  that  side.  This  is  true  also  of  all  wreath-pieces  having  butt-joints  falling  within  a  cir- 
cular plan,  but  these  straight  sides  are  corrected  in  the  hands  of  a  skilful  rail-worker,  who,  leaving  some 
over-wood — after  the  two  pieces  are  bolted  together — works  the  sides  plumb  with  the  proper-shaped  tools 
Fig.  2  shows  the  wreath-piece  with  the  concave  side  cut  away  plumb.  When  this  side  is  worked  plumb, 
a  gauge,  like  Fig.  8,  having  an  arm  provided  with  a  large  pencil,  may  be  used  to  mark  the  width  on 
the  convex  side  ;  next  the  top  is  shaped,  and  from  this  the  thickness  is  gauged.*  The  directions  here 
given  with  regard    to  this  wreath-piece  apply  generally  to  all,  but  particularly  to  the  following  : 

Plate  24,  Fig.  3,  Plate  25,  Fig.  6,  Plate  26,  Fig.  4,  Plate  27,  Fig.  4, 

"    32,    •'    3,  "    37-    "     4.  "    40,    "     6,  "    43.    "  3, 

"    44,     "     5,  "    46.    "     3,  "    54.    "  3- 

Figs.  3  and  4.  Squaring  a  Wreath-piece  from  a  Face-mould,  both  Tangents  of  which  are 
Inclined  either  on  a  Common  Inclination  or  on  Different  Inclinations. — The  angle  with  which  to  square 
a  Vv'realh-piece  at  the  butt-joint  is  the  inclination  of  the  face  of  the  plank  along  the  joint  given  by  the 
face-mould,  in  connection  with  a  line  on  the  joint — which  is  square  thiough  the  plank — that  coincides 
with  a  vertical  plane;   hence   it  is  commonly  understood  as  an   angle   giving  a   plumb-line    on   a  butt-joint 

To  Determine  the  Direction  in  which  to  Apply  the  Angles  for  Squaring  a  Wreath-piece  at  the 
Joints  : — Place  the  lower  end  of  the  wreath-piece  towards  you,  turn  the  upper  end  to  the  right  or  left 
to  suit  the  hand  of  the  stairs,  then  move  the  face-mould  up  a  few  inches  on  the  slide-line  as  at  Fig.  3, 
and  it  will  be  seen  that  as  the  centre  of  the  joint  J  is  carried  toward  K,  the  plumb-line  must  apply  in 
the  direction  K  M  passing  through  the  centre  L,  and  at  the  upper  end  the  other  tangent  will  be  moved 
towards  N,  showing  that  the  plumb-line  on  that  joint  must  lie  in  the  direction  N  P,  passing  through  the 
centre  0.     The  plank  edge  A   D,  bcvch  U  and  V,  show  at  once  the  correct  position  of  bevels. 

Another  Way  of  Deciding  the  Direction  in  which  to  Apply  Plumb-lines  on  Butt-joints  is  as 
Follows: — Holding  the  wreath-piece  as  befoi'e  directed,  cant  it  u\)  on  the  corner  F — a  prisition  it  must 
take — when  it  will  be  at  once  evident  that  the  plumb-line  must  apply  in  the  direction  K  M.  This  causes 
the  stock  of  the  bevel  to  lay  towards  the  convex  ;  then  at  the  upper  joint  reverse  the  stock  of  the 
bevel,  placing  it  towards  the  concave  as  shown  ;  also  the  slide-line  need  not  be  used,  but  the  tangent 
line  K  J  (squared  from  the  joint  at  K)  and  the  face-mould  tangent  moved  on  this  line  until  it  reaches 
the  ptiint   S  and   N   of  the   upper  end. 

To  Put  the  Face-mould  in  Position  on  the  Planes  of  the  Plank  so  that  its  Edges  will  Mark 
the  Plumb  Sides  of  the  Wreath-piece  : — Hold  the  wreath-piece  as  before  explained  ;  square  a  line  from 
the  joint  at  K,  K  J  indehnitely  ;  slide  ihe  face-mould  up  from  the  lower  end  along  the  slide-line  until 
J — tiie  centre  of  the  face-mould  joint — falls  on  the  line  K  J,  K  being  a  point  at  the  face  of  the  plank 
of  the  previously-applied  plumb  bevel  K  M  ;  then  again  when  tlie  face-uiould  is  in  this  position  its  tan- 
gent at  the  upper  end  will  touch  the  point  N  of  the  pluuib-line  N  P  on  the  upper  joint  ;  also  the  con- 
cave edge  of  the  face-mould  will  touch  S  of  the  plumb-line  S  T.  Apply  the  fiice-mould  to  the  other  side  of 
the  wreath-piece  on  the  slide-line,  keeping  the  joint  J  of  the  face-mould  as  much  below  the  joint  of  the 
wreath-piece  as  it  is  above  it  on  this  side. 

Fig.  4.  The  Wreath-piece  Shown  with  the  Concave  Side  Cut  Away  Plumb. — In  all  face-moulds 
of  this  character,  where  a  level  line  passes  through  tiie  centre  from  which  the  jilan  of  the  rail  is  described 
— the  minor  axis — there  is  a  place  where  the  plane  of  the  plank  is  level,  and  this  point  on  the  width  X  Y 
and  through  the  thickness  X  T  is  the  normal  place  in  a  wreath-piece,  and  where  the  over-wood  is  removed 
equally  parallel  to  the  faces  of  the  plank.  In  shaping  the  wreath-piece  the  centre  line  of  the  thickness  of 
rail  will  correctly  touch  R  and  U,  the  centre  at  the  joints,  and  S  the  centre  of  the  plank.  When  the  face- 
mould  is  in  position  to  plumb  the  sides  of  the  wreath-piece,  if  the  normal  place  is  marked  at  V  and  W,  the 
plumb-line  V  W  will  pass  through  the  centre  of  stuff  at  S,  and  give  the  direction  in  which  to  move  the 
round-faced  plane  in  working  the  sides  of  the  wreath-piece  plumb.  In  gauging  tlie  wreath-piece  to  a  width, 
the  long  arm  of  the  gauge  should  be  held  in  the  direction  of  the  plumb-line  W  V.  The  instruction,  g'ven  under 
the  head  of  Figs.  3  and  4  apply  particularly  to  the  following  : 

Plate  25,  Fig.  4,  Plate  26,  Fig.  6,  Plate  27,  Fig.  6,  Plate  28,  Fig.  4,  Plate  29,  Fig.  4, 


31.  " 

'     2  and  4, 

"    36,  ■ 

'  3, 

"    37,  ' 

'  7, 

"    38,  ' 

'  4, 

"    39.  ' 

'  4. 

40,  " 

'  3- 

"    41,  ' 

•  4. 

"42,  ' 

'     4  and  6, 

"    43.  ' 

'  5, 

"    44.  ' 

'  4, 

46,  " 

'  5, 

"    47,  ' 

'  4, 

"    50,  ' 

'  6, 

"    51,  ' 

'     3  and  5, 

"    52,  ' 

'  4- 

53,  " 

'  6. 

"    55,  " 

2,  4  and  7. 

Fig.  5.  Wreath-piece  from  a  Face-mould  over  a  Plan  of  Less  than  a  Quarter-circle,  the 
Position  of  A  K,  one  of  the  Tangents,  being  Level,  the  ether,  K  Z,  Inclined.— The  dotted  lines  show 
the  face-mould  as  placed  at  the  joint  D  to  plumb  the  sides  of  the  wreath  ;f  the  tangent  A  of  the  face-mould 
is  brought  to  C  on  the  other  side  of  the  wreath-piece.  The  sliding  of  face-moulds  of  this  character  is  always 
along  the  joint,  which  is  at  right  angles  to  the  level  tangent,  the  same  as  Fig.  i.  The  above  instructions 
apply  particularly  to  the  following: 

Plate  24,  Fig.  6,  "Plate  30,  Fig.  2  and  4,  Plate  32,  Fig.  6,  Plate  33,  Fig.  3,  Plate  48,  Fig.  4, 

"    49.    "     3.  "     50.    "     5  and  7,  "    53,     "     5  and  7,         "    55,    "  10. 


*  See  subject  of  Side  Moulds  as  treated  of  at  Pl.\te  No.  76. 

\  At  Fig.  5  and  the  list  given  in  tlie  applieation  of  the  angles  to  square  wreath-pieees  at  the  joints,  tlie  stock  of  the  bevel  at  both  joints  Hcs 
to-vards  tin  convex  side  of  tlte  ~'jreath-pieee  as  shoivn. 


PLATE  57. 


Fig.  I.  Plan  of  Stairs  Suitable  for  Wholesale  Stores. — These  stairs  from  the  first  to 
the  second  story  are  enclosed  with  panel-work  as  shown  by  the  elevation  Fk;  4.  The  door 
to  shut  off  communication  between  the  two  stories  is  often  placed  on  the  platform,  and  in 
that  case  the  platform  is  so  situated  that  the  door  trims  under  the  end  of  the  well-hole. 
Side-rails  are  hung  on  strong  ornamental  iron  brackets,  sometimes  on  both  sides  of  wide  flights. 
The  newel-posts  are  never  less  than  seven  inches,  and  those  at  the  top  of  the  flights  are 
continued  below  the  ceiling,  finishing  at  the  Invver  end   with   turned  work. 

Fig.  2.  Construction  of  Close  String  Paneled. — This  finish  of  string  is  used  in  the 
upper  flights  tiiat  are  furnished  wiih  hand-rail  and  balusters,  as  at  Fig  5  The  well-holes  of 
each  story  are  framed  shorter  than  the  run  of  the  flights  above,  so  that  each  flight  starts 
from  the  floor  below,  resting  directly  on   the  floor-beams. 

Fig.  3.  Panel-work. — By  this  plan  the  middle  muntins  A  are  wider  than  the  face-muntins 
D,  so  that  the  mouldings  may  be  nailed  free  from  the  panels,  allowing  the  latter  to  shrink 
without  disturbing  the  mouldings. 


PLATE  58. 

Fig.  I.  Plan  of  the  Landing-  Portion  of  a  Staircase  with  Square  Corner-pieces 
like   Small   Low-down   Newels  set   in  the  Angles  with  a  Continued  Hand-rail  Over. — 

This  plan  is  given  at  Plate  No.  6,  Fig.  6  L  and  M  are  3V'  square  angle-pieces  that  are 
brought  above  the  platform  and  floor  as  shown  in  connection  with  the  elevations  Figs  3  and 
4.  The  centres  of  balusters  A  and  B  are  equal  in  distance  to  A  and  Q.  The  square  angle 
at  the  corner-piece  L  is  turned  for  the  continued  hand-rail  one  quarter  C  D  V,  C  R  S,  using 
only  V  radius  from  the  face  of  the  corner-piece  E.  to  D,  then  from  the  joint  D  R  of  the  level 
quarter  another  is  taken  at  the  ramp  to  ease  over  to  this  level,  as  shown  at  the  elevation 
Fig  3,  Z  X  and  X  K. 

•  Fig.  2.  Design  and  Construction  of  Close  Front-string. — The  drawing  to  the  scale 
given  will  be  a  sufficient  explanation 

Figs.  3  and  4.  Elevation  of  Treads  and  Rises,  including  the  Square  Corner-piece 
Connecting  with   the  Platform,  the  String  and    the   Level   as  given  at  Plan  Fig.  i. — 

0  P  equals  0  P  of  Fig.  i.  The  face  P  E  of  the  corner-piece  L,  Fig,  i.  is  along  the  line  PEZ; 
Z  X  equals  D  E  of  Fig.  i;  Z  K  equals  one  inch;  the  joint  J  H  connects  with  D  R  of  Fig  i 
The  height  to  the  bottom  of  rail  Y  from  the  step  at  the  line  of  rise  W  is  2', i".  The  height 
from  floor  to  bottom  of  level  rail  SJ  is  2'.6".  C  D,  V  V  is  the  carriage-timber,  showing  its 
bearing  against  the  front  platform-timber  at  V  V.  F,  N  and  T  are  places  of  mortices  to  receive 
the  tenons  of  string.  The  baluster  at  B  is  intended  to  be  set  three  eighths  of  an  inch  into 
the  mill-plowed  hand-rail,  as  shown  at  the  section  A,  and  then  pieces  set  between  each  baluster, 
^"  thick,  thus  leaving  a  finished  panel  or  sinkage  of  -J",  the  depth  of  which  gives  a  much  better 
appearance  to  the  bottom  of  the  rail  than  when  flush. 


PLATE  59. 


F'-g.  I.  Plan  of  the  Upper  Portion  of  a  Quarter  Platform  Staircase  with  Square- 
moulded  Newels  set  in  the  Angles. — Tliis  plan  is  given  at  Plate  No.  7,  Fic.  5.  The  clotted 
lines  show  a  portion  of  the  rough  Iraming  of  the  jilatform. 

Fig.  2.  Design  Elevation  and  Details  of  Plan  Fig.  i. — Through  this  elevation  tiie 
lengths  of  the  angle  newels  and  the  connections  of  hand-rail,  balustrade  work,  strings,  etc.,  are 
obtained.  The  face  of  the  newel  marked  A  at  the  lower  end  showing  its  connections  is  face  A  at 
plan  Fig.  i  ;  also  the  face  of  the  newel  marked  E  at  the  lower  end  is  face  E  at  plan  Fig.  i. 

Fig.  3.  Laying  Out  the  Newel  connected  with  the  Platform,  the  Sides  of  which  are 
Lettered  on  the  Plan  Fig.  I,  A  B  C  D. — The  fom-  faces  of  this  newel  are  lettered  at  tiie  top  lo 
correspond  with  the  plan  Fig.  i.  J  K  is  the  total  length  of  the  newel-shaft  as  taken  from  J  K 
of  the  elevation  Fig.  2.  The  distances  marked  by  the  letters  J  L  M  N  indicate  principal  points 
of  measurement  taken  from  the  coi  responding  letters  of  the  A  side  of  newel,  Fig.  2  ;  and  the 
same  may  be  said  of  the  letters  0  P  Q  at  face  D  ;  the  sides  B  and  C  will  be  understood  by 
comparing  them   with  their  adjoining  sides   and   connections,   and    B  anrl   C  of  Fig.  i. 

Fig.  4.  Laying  out  the  Landing  Newel  the  Sides  of  which  are  Lettered  on  the 
Plan  Fig.  I,  E  F  G  H. — The  letters  at  the  top  of  these  sides  correspond  with  those  of  the  plan 
Fig.  I.  R  S  of  the  side  E  marks  the  total  length  of  this  newel-shaft,  and  T  U  V  the  prin- 
cipal points  of  measurement  lettered  the  same  at.  Fig.  2.  WXY  of  the  connection  at  the 
side  F  are  also  the  principal  points  of  measurement  taken  from  the  elevation  of  the  lauding 
newel.  Fig.  2.  The  sides  H  and  G  will  he  understood  by  examining  them  in  connection  with 
the  adjoining  sides  and   G   and    H    of   F'lc;.  i. 

Fig.  5.    Balustrade  Moulding  as  Shown  in  Place  at  the  Elevation  Fig.  2. 

Fig.  6.  Construction  of  Square  Newel-posts. — The  narrow  pieces  forming  the  sides  A 
and  B  of  tlie  newel-shaft  should  have  blocks  glued  to  tlie  inside  faces  at  the  edges — not 
more  than  one  foot  apart — wdien  the  glue  is  set  to  be  jointed  with  the  edges  square  from, 
the  face  ;  also  to  guard  against  the  joints  giving  way,  hard-wood  dowels  ought  to  be  set 
in  as  shown  at  suitable  intervals. 


Plate  No.59 


Plate  No.  61 


Plate  No. 62 


^3 


Plan  of  stairs  tuiniu;^  one-quarter  witli 
jiiutforin,  itnd  two  risers  curved  at  their 
front  ends  to  newels. --  The  plan  is  designed 
to  dispense  with  winders  and  give  a  com- 
fortable, easy  stairs  for  travel,  taking  but  a 
few  incliea  more  room  than  the  usual 
winders  that  are  required  to  make  a  quarter 
turn.  The  small  angle  uewcl  receives  the 
straight  rail  of  the  flight  uud  the  level 
hand-rail  at  the  landing,  as  shown,  by  the 
side  elevation.  This  mode  of  construction 
requires  no  twists  or  easements  ;  ilv  only 
turn  of  the  hand-rail  ie  the  level  quarter 
cylinder.    See  plate  5,  Fig.  7. 


Scale  1 


Plate  No. 64 


Plate  No.  66. 


Fig.  2. — Design  for  Spiral-turned  Newels 
and  Ralustcrs,  Braclceted  String;  Hand- 
rail with  Kainp  and  (3oose-neck. 


Elevation. 


Platform. 


Plan. 


Fi 


G.  3. 


Fig.  I. 


Figs.  3  and  4.— Platform  Stairs  with  Angle  Newels.  The 
bottom  riser  of  the  upper  flight  is  set  one  tread  from 
the  centre  of  newel  as  shown  on  the  plan,  FiG.  3,  and 
the  elevation,  FiG.  4;  if,  on  the  contrary,  the  bottom 
riser  referred  to  should  be  placed  at  the  centre  of  the 
newel,  it  would  then  be  necessary  to  make  that  newel 
one  rise  higher  from  the  platform  to 
receive  the  hand-rail  of  the  upper  flight. 
This  difference  in  the  height  of  the  two 
newels  from  the  same  platform  is  ob- 
jected to  by  some,  hence  these  sketchc 
and  explanations. 


FiG.  I. — Design  for  a  Turned  and  Carved  Newel,  Carved 
String  and  Balu.strade. 

 —  ..^^  I 


Plate  No.  67. 


Ancient  Staircase  at  Rouen,  France.— From  clie  '^American  Architect  and  Building  News. 


Plate  No.  68. 


Plate  No.  69. 


Plate  No.  70. 


PLATE  No.  71 


Scale    IxIn.  ^IFt. 


PLATE  73. 


Panelled  Soffit  of  a  Circular  Stairs  Showing  how  to  Work  Out  the  Twisted  or  Warped  Panels, 
Circular  Bands,  Radial  Bands  and  Mouldings  required. 

Fig.  I.  A  single  section  from  the  circular  plan  of  a  panelled  sofilit  embracing  three  threads.  This 
section  at  the  front  string  A  A  is  bounded  by  a  soffit  drop  moulding  marked  S  0  F.  M'  G — shown  at  the  cross- 
section  Fig.  2,  marked  C — and  a  circular  front  band,  at  cross-section  Fig.  2,  marked  B  ;  then  by  a  circular 
band  and  cornice  moulding  at  the  wall,  marked  at  the  cross-section  Fig.  2,  E  and  F  ;  further  there  are  two 
straight  ])arallel  radial  cross  bands,  the  centres  of  which  are  placed  along  the  line  of  the  first  and  fourth  risers. 
The  middle  band — at  cross-section  Fig.  2,  marked  D — divides  the  section  into  two  panels  of  equal  width. 

Fig.  3.  Elevation  of  Treads  and  Rises  at  the  Front  and  on  Line  of  Wall.— The  tread  B  A 
at  the  Iront  is  taken  along  the  convex  line  of  string  A  A,  and  the  tread  C  D  is  taken  at  the  wall,  which  is  the 
convex  line  of  the  cornice  moulding.  A  E  is'the  depth  of  string  that  may  be  required  ;  E  F  is  the  thickness  of 
bands.  At  the  place  of  fourth  rise  H  draw  H  G  parallel  to  rise  line  ;  prolong  B  1  to  J  ;  make  J  and  G  the 
centres  of  radial  bands,  which  as  shaded  show  the  ends  of  these  bands  in  their  position  as  given  at  plan  Fig. 
i;  prolong  the  rise  line  C  N  to  K  indefinitely  ;  from  M  parallel  to  rise  line  draw  M  L  indefinitely  ;  from  G  and 
J  draw  the  horizontal  lines  G  L  and  J  K  ;  connect  L  K,  and  parallel  to  L  K  draw  the  line  of  thickness  equal 
to  E  F  ;  then  L  and  K  are  the  centres  of  these  radial  bands,  as  shaded  in  position  at  line  of  wall.  Surround 
the  shaded  ends  of  bands  by  vertical  and  horizontal  lines  as  at  P  0,  Q  R,  T  S  and  V  U,  which  show  the  width 
and  thickness  of  wood  required  to  twist  these  radial  bands  to  the  warped  surface  of  the  soffit. 

Fig.  4.  Radial  Band  Showing  End  Sections  and  Exact  Length  Prepared  Ready  to 
Work. — At  the  elevation  Fig.  3  the  end  sections  as  enclosed  show  a  slight  difference  in  width  and  thick- 
ness. This  is  disregarded,  and  the  wood  got  out  of  ecjual  thickness  at  both  ends  ;  but  care  must  be  taken 
that  the  ends  are  laid  out  relatively,  just  the  same  as  at  the  top  and  sides  0  P,  0  R  and  T  V,  T  S  of  Fig.  3. 
0  T  must  be  the  exact  length  from  wall  line  to  convex  face  of  string  A  A,  plan  Fig.  i. 

Fig.  5.  Face-mould  for  Soffit  Moulding  under  Front  String  and  Band ;  Marked  at  Plan 
Fig.  I,  S  0  F.  M'G. — Beginning  at  plan  Fig.  i  make  the  perpendiculars  from  tangents  C  D  and  RG  each 
equal  in  height  one  and  a  half  rises  ;  connect  G  C  and  D  Z  ;  through  Z  draw  R  5  indefinitely  ;  on  C  as  centre 
with  C  G  as  radius  describe  an  arc  at  5;  from  S  at  right  angles  to  R  5  draw  C  S;  parallel  to  C  S  draw  FX 
and  J  M  B  ;  at  right  angles  to  C  Z  draw  M  N  and  X  0  ;  prolong  C  R  to  H  indefinitely  ;  on  R  as  centre  with 
R  E  as  radius  describe  the  arc  E  H  ;  connect  H  W  :  then  the  bevel  at  H  indicates  the  angle  that  will  square  this 
piece  0/ soffit  moulding  at  both  ends.  Now  at  Fig.  5  draw  the  line  L  P  ;  make  K  P,  K  L  each  equal  5  5  at  plan  ; 
draw  K  Q  at  right  angles  to  L  K  ;  make  K  Q  equal  S  C  at  plan  ;  connect  Q  P,  Q  L  ;  make  P  G  I  and  L  G  I 
each  equal  Z  0  N  at  plan  ;  make  G  D,  I  Y  H  each  equal  F  X  and  J  M  B  at  plan  ;  trace  the  curves  through  the 
points  thus  found.     Make  the  joints  at  right  angles  to  the  tangents. 

Fig.  6.  Face-mould  for  Panel  Marked  P  A  N  E  L  at  Plan  Fig.  I. — Beginning  at  plan,  all  the  lines 
and  angles  are  dotted  to  better  distinguish  the  panel  and  its  preparation  for  drawing  a  face-mould,  as  follows  :  Let 
J  and  V  i)e  the  centre  of  the  width  of  panel  at  the  ends  ;  connect  J  V  and  prolong  to  P  indefinitely  ;  at  right 
angles  to  2  V,  also  to  I  J, draw  the  tangents  V  K  and  J  K  ;  at  right  angles  to  V  K  draw  K  S  ;  make  K  S  and  J  0 
each  equal  one  and  a  half  rises  ;  connect  S  V  and  0  K  ;  through  K  at  right  angles  to  J  V  draw  A  A  ;  parallel 
to  A  A  draw  8,  8  and  2,  6  ;  at  right  angles  to  K  V  draw  4  Y  and  3  U  ;  prolong  K  V  to  Q  ;  on  K  as  centre  with 
K  0  as  radius  describe  an  arc  at  P  ;  prolong  8,  8  to  R  indefinitely  ;  prolong  V  2  to  R  ;  make  V  Q  equal  3  T  ; 
connect  Q  R  :  then  the  bevel  at  Q  indicates  the  angle  required  to  square  the  panel  at  both  ends.  Now  at  Fig.  6 
draw  the  line  S  S  ;  make  E  S,  E  S  each  equal  L  P  at  plan  ;  through  E  at  right  angles  to  S  S  draw  F  G  ;  make 
E  0  ecpial  L  K  at  plan  ;  connect  0  S,  0  S  ;  make  the  ends  S,  S  at  right  angles  to  the  tangents  S  0  ;  make 
S  Z  T  each  way  from  the  centre  equal  V  Y  U  at  plan  ;  through  Z  and  T  parallel  to  G  F  draw  R  N  and  K  L  ; 
make  Z  N,  Z  R  equal  4,  2,  4,  6  at  plan  ;  make  T  K,  T  L,  0  G,  0  equal  3,  8,  3,  8,  K  A,  K  A  of  plan  ;  through  S 
draw  R  U  ;  make  S  U  equal  S  R  ;  through  the  ]ioints  thus  found  trace  the  curves. 

Fig.  7.  Face-mould  for  Circular  Middle  Band. — Beginning  at  plan  Fig  i,  draw  the  line  Z  Z  E  ; 
make  tlie  tangents  Z  R,  Z  R  at  right  angles  to  Z  W,  Z  W; — W  is  the  centre  from  which  the  plan  was  described  ; 
— at  right  angles  to  Z  Z  from  R  draw  R  W  ;  make  the  perpendiculars  from  tangents  R  J  and  Z  1  each  one  and 
a  half  rises  ;  connect  J  Z  and  I  R  ;  parallel  to  R  W  draw  2  F  and  L  0  ;  at  right  angles  to  R  Z  draw  V  K  and 
OX;  on  R  as  centre  witli  R  1  as  radius  describe  the  arc  IE;  on  Z  as  centre  describe  the  arc  M  Y  ;  connect 
Y  W  :  then  the  bevel  at  Y  indicates  the  angle  that  ivill  square  the  middle  band  at  both  ends.  Now  at  FiG.  7  draw  the 
line  Z  Z  ;  make  ^1,^1  each  equal  W  E  at  the  plan  ;  draw  Y  0  at  right  angles  to  Y  Z  ;  make  Y  0  equal  W  R 
at  plan  ;  connect  0  Z,  0  Z  ;  make  the  joints  Z,  Z  at  right  angles  to  0  Z  ;  make  Z  D  C  ecpial  Z  X  K  at  plan  ; 
parallel  to  0  Y  draw  C  J  A  and  C  J  A  ;  make  C  J,  C  A  equal  V  F,  V  2  at  plan  ;  make  D  S  equal  0  L  at  plan  ; 
through  Z  draw  S  F  ;  make  Z  F  equal  Z  S,  etc. 

Fig.  8.  Face-mould  for  Cornice  Moulding. — Beginning  at  Plan  Fig.  i,  through  A  and  T  draw 
the  line  A  Q  ;  from  M  at  right  angles  Q  P  draw  M  P  ;  make  M  L  and  T  S  the  vertical  lines  from  tangents  M  T 
and  M  A  each  equal  one  and  a  half  rises  ;  connect  L  A  and  S  M  ;  prolong  M  T  to  R  ;  on  T  as  centre  describe 
the  arc  0  R  :  connect  R  with  W,  the  centre  of  plan  :  then  the  bevel  at  R  indicates  the  angle  required  to  square  the 
moulding  at  both  ends.  On  M  as  centre  with  M  S  as  radius  describe  tlie  arc  S  Q  ;  parallel  to  M  P  draw  H  I,  D  F 
and  C  V  ;  at  right  angles  to  M  A  draw  I  K,  F  G  and  V  B.  Now  at  Fig.  8  draw  the  line  C  B  ;  make  K  C,  K  B 
each  equal  P  Q  at  plan  ;  at  right  angles  to  K  C  draw  K  D  ;  make  K  D  equal  P  M  at  plan  ;  connect  D  B,  D  C  ; 
make  the  ends  B  and  C  at  right  angles  to  the  tangents  ;  make  B  0  M  L  and  C  0  M  L  equal  A  B  G  K  at  plan  ; 
parallel  to  D  K  draw  L  U,  M  T  and  0  R  ;  measure  the  points  as  before  from  tangents  at  L  M  and  0,  etc.  The 
shaded  end  sections  show  the  squaring  of  the  moulding  at  both  ends.  No  face  moulds  are  drawn  for  the 
front  and  wall  bands,  as  the  proceeding  is  precisely  the  same  as  for  Fig.  7,  the  middle  band.  All  the  above 
face-moulds  are  alike  in  character,  and  are  explained  in  detail  at  Plate  15.  See  Plate  84,  Figs.  4,  5  and  6, 
for  the  same  case  of  face-mould  and  its  management  in  squaring.    See  also  Plate  56. 


PLATE  74. 


Another  and  Third  Method  of  Treating  Hand-rail  Over  a  Large  Cylinder,  Two  Ways  o* 

WHICH  are  Given  at  Plate  24. 

Fig.  I.  Plan  of  15"  Cylinder,  the  Hand  rail  and  Elevation  of  Step  and  Rises  to  Floor  at 
Top  Landing. — Let  J,  the  centre  of  the  hand-rail,  be  fixed  over  the  phm  D  at  its  recjuired  height  above  the 
floor  ;  with  J  as  centre  describe  a  circle  equal  in  diameter  to  the  thickness  of  the  plank  out  of  which  the 
wreath-piece  is  to  be  worked  ;  draw  the  bottom  line  of  rail  through  the  centre  places  of  short  balusters  X  X. 
At  K,  four  inches  below  the  chord,  for  straight  wood,  draw  the  joint  line  K  8  at  right  angles  to  X  X  ;  from  K 
draw  the  line  K  A,  touching  the  circle  at  A  ;  parallel  to  K  A,  touching  the  circle  at  M,  draw  8  M  ;  make  K  L 
equal  the  thickness  of  rail  ;  from  L  draw  a  line  parallel  to  X  X.  The  bevel  at  8  indicates  the  angle  to  be  used 
for  the  joint  through  the  thickness  of  the  plank.  The  bevel  at  A — instead  of  the  pitch-board — is  used  to 
square  the  wreath-yjiece  as  shown  at  Fig.  2,  joint  D  or  J. 

Fig.  2.    Face-mould. — The  measurements  A  1  B  2  C,  etc.,  correspond  with  and  are  taken  from  Fig.  1. 

Fig.  3.  Plan  of  Setting  Off  a  Newel. — The  rail  may  be  shaped  by  using  an  ordinary  easement  cut 
out  of  plank  as  thick  as  B  C,  so  that  the  side  curves  F  A  and  E  G  may  be  formed.  Tlie  stair-string  need  not 
be  curved  or  bent  ;  only  let  the  end  of  the  bottom  step  project  sufficient  to  suit  the  ])lan  curve. 

Fig.  4.  Elevation  of  Steps  and  Rises  from  Plan  Fig.  3. — Let  A  B  equal  A  B  of  Fig.  3.  B  C  is 
six  inches,  and  may  be  more  or  less  as  recpiired.  C  E  e(|uals  C  E  of  Fig.  3.  The  bottom  of  the  rail  rests  at 
X  X,  the  centre  of  the  places  of  short  balusters. 

Fig.  5.  Case  of  Hand-rail  Like  that  Given  at  Plate  16,  Fig.  3,  with  this  Difference. — In  this 
case  the  level  line  U  L  common  to  both  planes  occurs  at  right  angles  to  the  tangent  A  B,  for  which  reason  the 
sides  of  this  wreath-piece  over  joint  A  are  at  right  angles  to  the  face  of  plank. 

Fig.  6.  Face-mould. — This  face-mould  is  measured  from  plan  Fig.  5.  The  squaring  of  the  wreath- 
piece  at  the  joints  shown,  the  sliding  of  the  face-mould  as  by  dotted  lines  in  position  to  plumb  the  sides  of  the 
twist,  are  all  given  without  further  explanation  than  the  preliminary  statements  and  the  drawings  themselves 
afford  ;  this  being  thought  ample  in  connection  with  the  full  details  given  at  Plate  16  and  repeated  at  other 
Plates. 


PLATE  75. 


Fig-.  I.  Plan  of  Stairs  Starting  which  Avoids  the  Old-fashioned  Angular  Winders  by 
Curving  Risers,  and  thus  Secures  Parallel  Steps  and  a  Roomy  Platform.    (See  also  Plate  46.) 

— Fig.  2.  Elevation  from  plan  Fig.  i.  Let  the  bottom  of  the  rail  above  and  below  rest  at  the  centre  of 
short  balusters  X  X  and  X  X  ;  draw  a  line  the  thickness  of  the  rail  above  X  X  as  shown.  F'ix  the  places  of 
chord  lines  or  commencement  of  cylinders  measured  from  plan  Fig.  i  at  C,  5,  J  and  M.  Set  off  the  length 
of  plan  tangent  at  C  B,  E  D,  J  G  and  M  L.  Let  A  be  the  centre  of  rail  ;  parallel  to  X  X  draw  A  B  ;  make 
2  A  2!"  for  straight  wood  ;  at  B  draw  B  C  parallel  to  tread  line.  From  4,  wliich  is  a  fixed  point,  draw  the 
line  4  F,  raising  or  lowering  the  point  F  to  suit  ;  make  E  F  2^"  for  straight  wood,  and  make  the  joint  of  ramp 
F  at  right  angles  to  4  F.  At  H,  the  centre  of  the  rail,  draw  H  G  parallel  to  X  X  ;  at  G  draw  G  J  parallel  to 
tread-line  ;  from  6,  which  is  a  fixed  point,  draw  the  line  5  M,  raising  or  lowering  the  point  M  at  pleasure  ; 
make  M  0  2I"  for  straight  wood,  and  make  the  joint  of  the  level  easement  0  at  right  angles  to  0  6  ;  from  the 
centre  P  describe  the  curve  of  easement  as  sliown. 

Fig-  3-  Plan  of  Rail  and  Tangents  Taken  from  Fig.  i  to  be  Prepared  for  Drawing 
the  Face-mould. — Let  D  4  equal  the  same  at  Fig.  2  ;  connect  4  E  ;  make  D  B  and  Z  C  equal  D  4  :  make 
C  2  equal  the  same  at  Fig.  2  ;  make  D  K  equal  C  2  ;  draw  K  A  parallel  to  D  E  ;  draw  A  0  parallel  to  K  D  ; 
from  0  draw  0  T,  which  is  the  level  line  common  to  both  planes  ;  parallel  to  0  T  draw  the  other  measuring 
lines  required  ;  at  right  angles  to  0  T  draw  E  P  and  Z  Q  ;  on  D  as  centre  with  4  E  as  radius  describe  an  arc 
at  P  ;  again  on  D  as  centre  with  B  2  as  radius  describe  an  arc  at  Q  ;  connect  P  Q.  To  find  the  angle  that 
will  square  the  wreath  over  the  joint  Z,  make  Z  Y  equal  C  L  ;  connect  Y  W:  then  the  bevel  at  Y  will  give  the 
angle  required.  To  find  the  angle  that  will  square  the  wreath  over  the  joint  E,  make  E  N  equal  0  X  ;  connect 
N  T  :  then  the  bevel  at  N  indicates  the  required  angle. 

Fig.  4.  Face-mould  for  Wreath-piece  Over  Both  Quarter  Cylinders. — The  corresponding 
letters  with  Fig.  3  show  the  measurements  by  which  to  draw  tlie  face-mould.  The  end  sections  together 
with  the  dotted  lines  of  face-mould  in  position  show  the  squaring  of  the  wreath  ])iece.  A  face-mould  like  this 
is  treated  in  full  detail  at  Plate  12.  Also  a  face-mould  of  this  character  will  be  found  at  Plate  27,  Fig.  6  ; 
Plate  41,  Fig.  4  ;  Plate  42,  Fig.  4  ;  Plate  44,  Fig.  4  ;  Plate  46,  Fig.  5.  This  case  of  hand-rail  may  be 
treated  differently  if  thought  desirable,  or  as  shown  by  the  dotted  lines,  as  follows  :  From  4,  Fig.  2,  draw  a 
straight  line  to  K  below  ;  then  K  is  at  the  base  of  the  upper  height  0  0,  and  the  lower  height  Z  Z  is  made  the 
same  as  0  0.  This  arrangement  of  the  line  4  K  and  the  equal  heights  0  0  and  Z  Z  would  give  one  face- 
mould  at  the  first  quarter  of  a  common  pitch,  and  one  above  of  two  different  inclinations  the  same  as  Fig.  4, 
but  not  quite  so  extreme  a  case  as  th.nt.  Then  the  ramp  is  got  rid  of,  and  the  rail  altogether  would  have  a  very 
easy  and  agreeable  outline  ;  but  at  the  fourth  rise  up  it  will  be  about  3"  too  high.  Would  this  variation  be  as 
objectionable  as  in  the  case  of  newels  in  place  of  quarter  cylinders  where  the  rail  cuts  straight  against  the 
newel,  as  it  does  in  such  open-newel  stairs,  causing  the  hand  to  feel  along  the  sides  of  the  newel  a  foot  or 
more  until  able  to  grasp  the  connecting  rail  above  ?  Even  if  the  rail  connections  with  the  newels  are  made 
by  ramp  and  knee — see  Plate  60  — it  is  possibly  even  then  far  more  awkward  than  such  slight  variations  in 
heii^hts  as  suggested  for  this  other  and  different  treatment  of  the  continued  rail.    See  Plate  46. 


Plate  No.  75 


Plate  No.  76 


PLATE  76 


Side-moulds  continued. — Also  How  to  Cut  Large  Square-top  Balusters,  Stone  or  Wood,  to 
AN  Exact  Length  ;  the  Tops  to  Fit  the  Warped  Bottom  Surface  of  Wreathed  Hand- 
railing. 

Fig.  I.  Plan  of  Semicircular  Wreath  to  be  Prepared  for  the  Measurement  and  Unfold- 
ment  of  Side-moulds. — 5  K  8  6  L  are  plain  tangents  to  centre  line,  5  8  L  ;  let  the  three  heights  raised  over 
these  tangents  equal  6  A  of  a  common  inclination,  A  D  4  5  ;  divide  the  circular  centre  line  into  any  number 
of  equal  parts,  say  eight  ;  draw  radial  lines  from  the  centre  S  through  each  of  these  divisions,  touching  the 
concave  V  T  W  and  the  convex  X  Y  U  ;  from  K  draw  the  level  line  K  S  ;  parallel  to  K  S  draw  0  9  and  Q  J  ; 
parallel  to  6  A  draw  M  1 ,  N  B,  P  2,  8  C,  and  J  3  ;  parallel  to  K  4  draw  9  E. 

Fig.  2.  Unfoldment  of  the  Concave  Side-mould.— Mark  on  the  line  5,  6  tlie  eight  divisions  at 
V  T  W  of  Fig.  i  ;  set  up  the  heights  6  A,  F  1,  etc.,  taken  as  indicated  by  the  corresponding  letters  at  Fig.  i  ; 
make  5  T  equal  K  4  and  T  P  equal  5  K  of  Fig.  i  ;  5  Z  is  straight  wood  ;  A  S  is  also  tor  straight  wood  ;  the 
joint  Z  is  made  at  right  angles  to  5  P  ;  the  joint  at  C  is  the  centre  of  the  upper  and  lower  wreath-pieces,  and 
is  made  at  right  angles  to  5  P  ;  at  A  1  B  2,  etc.,  describe  circles  equal  in  diameter  to  the  thickness  of  rail. 
Trace  lines  touching  the  circles  at  opposite  sides,  and  this  completes  the  side-mould. 

Fig.  3.  Unfoldment  of  the  Convex  Side-mould. — The  heights  by  which  points  for  tracing  the 
unfoldment  (also  the  joints)  are  as  shown,  the  same  as  at  Fig.  2  ;  the  only  difference  is  that  the  eight  divi- 
sions of  the  convex  X  Y  U,  Fig.  i,  are  marked  along  the  line  R  W.  A  slight  change  in  form  as  made  from 
the  two  points  above  W,  I  would  advise  wherever  in  a  side-mould  it  may  seem  desirable.  These  side-moulds 
are  to  be  cut  apart  at  joint  C  ;  they  apply  to  the  ])luml)ed  sides  of  the  wreath-pieces. 

Fig.  4.  Plan— to  be  Prepared  for  the  Unfoldment  of  Side-moulds— of  a  Wreath-piece,  One 
of  its  Tangents  a  Level  Line,  Producing  a  Twist  of  Double  Curvature,  the  Upper  Curve 
Easing  to  a  Level. — Divide  the  centre  line  A  B  into  say  four  equal  parts,  and  through  these  draw  radial 
lines  from  the  centre  S,  touching  the  convex  and  concave  P  Q  and  N  0  ;  through  the  points  E,  F,  G  draw  lines 
parallel  to  A  C  and  touching  the  tangent  B  C  ;  let  B  D  be  the  height  raised  over  the  tangent  B  C  ;  draw  the 
perpendiculars  H  M,  I  L  and  J  K.  This  plan  is  taken  from  Fig.  2,  Plate  84,  it  being  the  first  wreath-piece 
joining  the  newel  of  the  stone  circular  stairs  given  at  Plate  81. 

Fig.  5.  Unfoldment  of  the  Convex  Side-mould. — Mark  on  the  line  B  Q  the  four  parts  from  P  to 
Q  of  Fig-  4.  Place  the  heights  B  D,  H  M,  etc.,  taken  as  indicated  by  the  corresponding  letters  at  Fig.  4; 
make  D  I  W  equal  C  B  D  of  Fig.  4  ;  make  the  joint  D  at  right  angles  to  D  W  ;  make  D  0  equal  5,  6  of  Plate 
8i  ;  make  0  X  6i-",  or  equal  one  rise  ;  then  X  D  is  the  inclination  of  the  convex  side  of  the  hand-rail,  which 
the  upper  and  lower  curves  of  this  side-mould  mast  tangent  as  shown.  In  this  case  it  happens  that  the  angle 
0  X  D  the  same  as  the  angle  I  W  D.  At  D,  M,  etc.,  describe  circles  in  diameter  equal  to  the  thickness  of 
rail,  and  touching  these  trace  the  upper  and  lower  curve  lines  of  the  completed  convex  side-mould. 

Fig.  6.  Unfoldment  of  the  Concave  Side-mould  from  Plan  Fig.  4. — Mark  on  the  line  Z  Y  the 
four  parts  taken  from  N,  0,  Fig.  4  ;  the  heights  are  the  same  as  D  M  L,  etc.  ;  of  Fig.  5  :  D  I  W  is  the  same  as 
at  Fig.  5  ;  tlie  joint  D  is  at  right  angles  to  D  W  ;  D  0  is  equal  to  2,  4,  of  Plate  81  ;  0  X  equals  one  rise  as  at 
P'iG.  5  ;  X  D  is  the  inclination  of  the  concave  side  of  rail,  which  the  upper  and  lower  curves  of  this  side- 
mould  must  tangent  as  shown. 

Fig.  7.  Plan  of  Wreath-piece  same  as  Fig.  4. — In  this  case,  however,  both  tangents  are  inclined 
so  as  to  get  a  distance  lower  down  equal  to  A  B,  and  force  an  easement  to  the  lower  level  in  the  thickness  of 
stone  or  wood,  as  shown  at  Figs.  8  and  g.  Divide  the  centre  line  C  D  into,  say,  four  equal  parts,  and  draw 
radial  lines  from  centre  S  ;  make  C  F  equal  B  D  of  Fig.  4.  The  face-mould  for  this  plan  and  its  application 
is  given  at  Plate  ii,  through  Figs.  7,  8,  9  and  10. 

Fig.  8.  Concave  Side-mould  from  Plan  Fig.  7. — Mark  on  the  line  E  G  the  four  parts  at  E  G  of 
Fig.  7  ;  make  G  D  equal  the  two  heights  A  B  and  C  F  of  Fig.  7  ;  make  D  I  W  equal  A  C  F  of  Fig.  7  ;  let 
D  0  X  equal  the  same  at  Fig.  6  ;  make  the  joint  D  at  right  angles  to  D  W  ;  sketch  the  curve  line  E  D  to 
tangent  the  line  D  X,  and  at  the  points  of  intersection  with  the  perpendiculars  describe  circles  equal  in 
diameter  to  the  thickness  of  rail  ;  then  trace  the  upper  and  lower  curved  edges  touching  the  circles. 

Fig.  9.  Convex  Side-mould  from  Plan  Fig.  7. — Mark  on  the  line  J  H  the  four  parts  J  H  at  Fig.  7. 
The  joint  D  and  the  heights  and  D  I  W  are  the  same  as  at  Fig.  7.  D  0  X,  the  inclination  of  the  convex  side 
of  the  rail,  is  the  same  as  at  Fig.  5.  These  side-moulds,  as  stated  above  of  the  plan,  apply  to  Figs.  7,  8,  9 
and  10  of  Plate  ii.  At  Plates  20  and  21  examples  of  the  unfoldment  of  the  centre  line  of  wreath  and 
wreath-pieces  are  given.    Read  the  remarks  at  the  beginning  of  Plate  20. 

Fig.  10.  To  Cut  Very  Large  Stone  or  Wood  Square  Top  Balusters  to  an  Exact  Length  : 
the  Tops  to  Fit  the  Warped  Bottom  Surface  of  Wreathed  Hand-railing. — Set  up  treads  and 
rises,  the  tread  to  equal  3,  3  at  the  centre  line  of  rail,  Plate  81.  Set  the  baluster  on  the  step  as  required  ; 
make  0  K  the  height  ;  at  K  draw  K  P  at  right  angles  to  0  K  ;  make  K  S  equal  0  0  at  the  concave  face  of 
balusters,  Plate  81  ;  draw  S  U  perpendicular  to  K  P  and  e(iual  to  one  rise  ;  through  K  draw  the  line  U  X, 
which  will  be  the  angle  and  length  of  baluster  facing  the  concave  side  of  rail.  For  the  side  of  baluster  facing 
the  convex  side  of  rail,  make  K  P  equal  X  X  at  Plate  81  ;  make  P  T  equal  S  U  ;  through  K  draw  the  line  T  K  Z. 
The  central  point  K  must  be  squared  through  to  the  opposite  face  of  baluster,  and  the  angle  T  K  Z  made  to 
pass  through  K.    On  the  other  two  faces  of  baluster  draw  lines  connecting  those  made  through  K. 


PLATE  76. 


The  Use  of  Side-moulds  in  Hand-railing. 

Peter  Nicholson,  the  eminent  practical  English  mathematician,  who  in  the  year  1792,  first  applied 
geometry  intelligently  and  correctly,  as  far  as  his  discoveries  carried  him,  to  the  use  and  requirements  of 
hand-railing,  gave  what  he  called  "  falling-moulds  " — patterns  of  a  width  equal  to  the  thickness  of  rail,  made 
to  fold  around  the  convex  or  concave  sides  of  wreaths,  or  wreath-pieces,  by  which  to  mark  the  shape  or  curves 
of  the  upper  and  under  surfaces  of  wreaths.  I  call  these  patterns  more  properly,  considering  their  use  and 
application,  side-moulds.  Mr.  Nicholson  did  not  work  from  the  centre  of  any  form  of  curved  hand-railing, 
nor  did  he  make  use  of  tangents  from  which  we  make  joints  across  the  width  and  through  the  thickness  of 
wreath-pieces,  fixing  also  by  their  adoption  the  exact  place,  practically  the  most  useful,  for  the  three  points 
controlling  the  inclination  of  the  plank.  This  author's  "  falling-mould  "  is  produced  by  the  simple  stretch 
out  of  convex  or  concave,  the  height,  straight  connecting  lines  and  curves  as  required  at  pleasure.  Our  side- 
moulds  are  unfolded  from  the  centre  of  the  wreath — that  is,  by  central  points  taken  at  the  centre  of  the  thick- 
ness and  the  centre  of  the  width;  and  thus  we  get  the  exact  mathematical  centres  and  resultant  geometrical 
curves,  which  we  are  at  liberty  to  alter  or  not  as  conditions  may  require.  In  this  city  no  stair-builders  working 
hand-rail  at  this  time  make  use  of  "  falling-moulds,"  or  side-moulds.  They  seem  to  be  content  with  the  results 
of  trial  practice,  and  the  experience  it  affords  in  a  general  way  of  the  curves  required  in  shaping  wreaths,  and 
this  experience  as  a  speciality  leads  to  many  men  working  only  at  hand-railing.  Certainly  no  man  who  has 
not  had  considerable  practice  can  take  up  a  twist  and  shape  it  correctly  without  being  overlooked  by  some 
skilful  workman  to  prevent  his  spoiling  it  ;  thus  taking  the  time  of  two  on  what  ought  to  be  one  man's  work. 
It  is  just  so  with  an  apprentice  who  is  too  often  wholly  kept  from  learning  how  to  do  this  interesting  and  use- 
ful kind  of  work  because  he  requires  so  much  time  and  attention.  Side-moulds  once  well  understood  take 
little  time  to  make  and  apply,  and  by  their  use  they  insure  absolutely  correct  and  graceful  curves  at  once 
without  the  necessity  of  spending  time  eyeing  the  twist  over  and  over  again,  or  leaving  overwood  on  each 
piece  ;  so  that  when  the  complete  wreath  is  bolted  together  there  is  more  eyeing,  shaping,  and  time  taken  in 
perfecting  the  curve,  and  very  likely  another  man  invited  to  spend  his  time  to  decide.  With  the  use  of  side- 
moulds  each  separate  twist-piece  may  be  worked  exactly  to  the  lines,  being  sure  of  correct  shape  and  thick- 
ness— no  "  little  too  thick  here,"  or  "  a  little  too  thin  there  "  ;  so,  too,  when  the  wreath-pieces  are  bolted 
together  there  will  be  no  more  finishing  at  the  joints  than  there  would  be  with  joinings  of  straight  pieces  of  rail. 
The  perfection  of  thickness  and  the  shape  of  the  upper  and  under  surfaces  of  wreaths  is  as  much  under 
the  control  of  the  draughtsman  in  making  correct  side-moulds  as  the  shape  of  the  sides  of  a  wreath  and  the 
width  of  the  rails  is  by  correct  face-moulds.  Side-moulds  must  be  got  out  of  some  flexible  material,  such  as 
straw-board  or  thin  sheet-zinc. 


PLATE  77. 


The  plan  here  presented  and  the  treatments  of  hand-rail  and  side-moulds  is  given  as  a  modified,  simpler 
form  than  that  given  at  Plate  5,  Fig.  6,  which  is  treated  at  Plate  26.  This  plan  takes  34"  more  run 
than  the  plan  Plate  26,  and  it  takes  ii^"  more  run  than  the  plan  with  five  winders,  which  is  shown  in 
comparison  alongside  the  Fig.  6  above  mentioned.  The  hand-rail  by  this  plan  of  stairs  is  executed  without 
the  long  ramp  required  by  the  plan  at  Plate  26.  It  seems  desirable  to  avoid  winders  if  comfortable,  square 
platforms  and  parallel  steps  can  be  planned  to  land  as  required  with  so  little  additional  space  as  here  shown. 
To  do  this  in  many  cases  it  might  even  be  better  to  raise  the  height  of  rises  a  little,  diminishing  their  number, 
or  contract  the  tread  some,  maybe  do  both,  rather  than  amble  up  and  down  those  really  awkward  and  to 
many  people  dangerously  huddled  steppings  on  tapering,  angular  winders. 

Fig.  I.  Plan  of  Quarter  Turn  Platform  Stairs  with  6"  Cylinder,  One  Curved  Step  and  Two 
Rises  Above  Platform. — The  tangents  to  centre  line  of  rail  are  2  A  D  X  3  ;  the  heights,  etc.,  raised  over 
tangents  is  taken  from  elevation,  as  explained  further  on. 

Fig.  2.  Elevation  of  Treads  and  Rises  Shown  at  Plan  Fig.  i. — Draw  the  plan  tangents  A  B, 
C  D  and  E  F  ;  parallel  to  the  rise  lines  draw  DEL  and  F  M  ;  let  the  bottom  of  the  rail  rest  on  X  X,  the 
centres  of  short  balusters,  and  set  off  the  thickness  of  rail  ;  parallel  to  X  X  draw  the  centre  line  of  rail  S  C  ; 
make  A  S  equal  the  straight  to  be  left  on  lower  twist ;  at  Fig.  i  make  A  M  and  D  Y  each  equal  D  E  and  F  M 
of  Fig.  2  ;  connect  M  D  and  Y  X  ;  make  X  E  equal  B  C  of  Fig.  2  ;  connect  E  3  ;  again  at  Fig.  i  make  D  K 
equal  X  E  ;  draw  K  N  at  right  angles  to  D  Y,  and  parallel  to  D  Y  draw  N  P  ;  connect  P  X  ;  parallel  to  P  X  from 
the  centre  of  baluster  0  draw  0  I,  and  from  Q  draw  Q  H  also  parallel  to  P  X  ;  from  R  parallel  to  A  M  draw 
R  B  F  ;  at  Fig.  2  make  C  J  equal  Q  D  of  Fig.  i  ;  parallel  to  D  E  draw  J  K  ;  make  J  K  equal  D  Y  of  Fig.  i  ; 
make  0  Z  equal  I  Z  of  Fig.  1  ;  make  C  Q  equal  H  L  of  Fig.  i  ;  make  E  L  equal  B  F  of  Fig.  i  ;  through 
A  Z  Q  K  L  M  trace  the  centre  curve,  and  from  the  points  A  Z  Q,  etc.,  draw  circles,  the  diameters  of  which  equal 
the  thickness  of  rail,  and  touching  these  circles  trace  the  edges  of  the  unfolded  central  section  of  rail.  M  P 
is  the  straight  allowod  on  the  upper  wreath-piece.  Side-moulds  for  this  case  may  be  drawn  readily  from 
previous  explanations;  also  from  tliose  that  will  follow  for  Figs.  8  and  9. 

Fig.  3.  Plan  of  Rail  taken  from  Fig.  i  to  be  Prepared  for  Drawing  the  Face-moulds. — Let 
the  heights  A  M,  D  Y  and  5  H  ecpial  F  M,  D  E  and  B  C  of  Fig.  2  ;  make  D  E  equal  5  H  ;  draw  E  P  at  right 
angles  to  D  Y  ;  parallel  to  D  Y  draw  PZ  8  ;  connect  Z  5,  the  governing  level  line  common  to  both  planes  ; 
parallel  to  Z  5  draw  F  B  B,  R  0,  U  C  and  Q  1  ;  parallel  to  D  Y  draw  V  3,  T  4,  B  I,  0  J  and  C  K  ;  at  right 
angles  to  Z  5  draw  D  W  and  Q  9  ;  on  5  as  centre  with  5  Y  as  radius  describe  the  arc  Y  W  ;  with  H  Q  as  radius 
on  5  as  centre  describe  the  arc  at  9  ;  connect  9  W.  To  find  the  angle  required  to  square  the  lower  wreath-piece  over 
joint  D,  make  D  2  equal  D  K  ;  connect  2  6  :  then  the  bevel  at  2  indicates  the  angle  sought.  The  angle  to 
square  the  wreath-piece  over  joint  Q  is  found  by  making  Z  8  equal  5  L;  connect  8  Q  ;  then  the  bevel  at  8  shows 
the  angle  required.     The  angle  for  squaring  the  11  ['per  7i<reath-piece  is  indicated  by  the  bevel  at  M. 

Fig.  4.  To  Drav/  the  Face-mould  for  the  Upper  Wreath-piece. — Make  M  3,  4  D  equal  M  3,  4  D 
of  Fig.  3  ;  draw  M  P,  3  K,  4  Z  T  and  S  D  R  at  right  angles  to  M  D  ;  make  M  G  etpial  A  G  of  Fig.  3  ;  make 
G  P  equal  the  straight  allowed  at  M  P,  Fig.  2  ;  through  G  draw  0  R  jjarallel  to  M  D  ;  make  D  S,  D  S  each 
equal  V  G  of  FiG.  3  ;  make  4  Z  equal  X  X  and  4  T  equal  X  T  of  Fig.  3  ;  make  G  0  and  G  V  each  equal  M  3  ; 
parallel  to  M  D  draw  J  P  K  ;  trace  the  curves  ST  V  and  S  Z  M  0  to  complete  the  face-mould.  The  angle  to 
square  the  wreath-piece  at  joint  P  is  taken  from  M,  Fig.  3. 

Fig.  5.  To  Draw  the  Face-mould  for  the  Lower  Wreath-piece. — Make  9  N  W  equal  9  N  W  of 
Fig.  3  ;  make  9,  5  equal  G  H  of  Fig.  3  ;  make  W  5  equal  5  Y  of  Fig.  3  ;  make  9  S  equal  A  S  of  Fig.  2  ;  make 
the  joint  at  S  at  right  angles  to  S  5,  and  the  joint  W  at  right  angles  to  W  5  ;  make  W  I  J  K  equal  Y  I  J  K  of 
Fig.  3  ;  connect  5  N  ;  parallel  to  5  N  draw  9  Z,  K  F,  J  R  and  I  T  B  ;  make  K  F  equal  C  U,  9  Z  equal  Q  1 ,  J  R 
equal  0  R,  and  I  B,  I  T  equal  B  B,  B  F,  all  of  Fig.  3  ;  through  W  draw  T  P  ;  make  W  P  equal  T  W  ;  through 
9  draw  F  L  ;  make  9  L  equal  9  F  ;  through  TRF,  PB5ZL  trace  the  curved  edges  of  the  face-mould.  The 
slide  line  is  at  right  angles  to  5  N.  The  angle  for  squaring  the  wreath  at  joint  W  is  shown  by  the  bevel  at  2, 
Fig.  3,  and  for  squaring  the  wreath  at  joint  S,  the  bevel  at  8,  Fig.  3.  The  above  face-mould  is  given  in  full 
details  at  Plate  12.  See  also  Fig.  6,  Plate  27.  There  is  one  palpable  defect  in  the  above  natural  treat- 
ment of  the  hand-rail  over  this  plan,  to  which  attention  is  invited,  and  that  is  at  Q,  as  may  be  seen  on  the 
line  C  Q,  Fig.  2,  where  the  rail  is  3!"  low.  If  this  is  thought  of  sufficient  importance  it  may  be  corrected  by 
another  and  different  treatment,  as  given  at  the  elevation  Fig.  6  and  in  connection  Figs.  7,  8  and  9.  Fig.  6. 
Set  up  an  elevation  same  as  Fig.  2  ;  also  set  off  the  plan  tangents  as  before  ;  M,  as  at  Fig.  2,  becomes  a  fixed 
point  ;  from  M  draw  the  line  M  S  at  pleasure,  raising  it  at  the  chord  A  to  suit  ;  through  the  centres  of  short 
balusters  draw  the  bottom  line  of  rail  X  X  ;  draw  S  0  at  the  centre  of  rail  and  parallel  to  X  X  ;  the  joint  T  V 
is  at  right  angles  to  the  tangent  A  C,  and  will  be  the  joint  of  the  wreath-piece,  as  worked,  and  when  squared 
up  the  last  thing  to  do  is  to  cut  away  the  material  V  Z  T,  leaving  the  required  joint  T  Z  at  right  angles  to  X  X  ; 
A  S  is  the  straight  allowed  ;  the  centre  joint  R  is  at  right  angles  to  the  tangent  C  E. 

Fig.  7.  Plan  of  Rail  from  which  to  Unfold  the  Central  Section  Fig.  6  and  Figs.  8  and  9 
Side-moulds. — Divide  the  centre  line  A  L  K  into,  say,  six  equal  parts  ;  draw  lines  from  the  centre  Y  through 
each  of  these  divisions  ;  connect  8  Y  ;  parallel  to  8  Y  draw  P  W  and  0  X  ;  make  4  W  equal  the  three  heights 
B  C,  D  E  and  F  M  of  Fig.  6  ;  parallel  to  4  M  draw  H  J,  P  Q,  L  R,  W  T  and  8,  8  ;  parallel  to  4,  8  draw  X  G. 
On  the  line  A  K,  Fig.  6,  mark  the  six  parts  at  centre  line  A  L  K  of  Fig.  7  ;  make  the  heights  K  M,  0  J,  P  Q, 
etc.,  equal  4  M,  H  J,  P  Q,  etc.,  of  Fig.  7  ;  through  the  points  thus  found  describe  circles  equal  in  diameter  to 
thickness  of  rail ;  trace  the  upper  and  lower  curves,  touching  the  circles,  except  the  one  at  A,  from  which  a 
slight  departure  is  made  to  produce  the  required  curve  at  that  point. 

Figs.  8  and  9.  Side-moulds  from  the  Prepared  Plan  Fig.  7.  The  heights  are  the  same  in  both 
of  these  side-moulds,  as  taken  from  Fig.  7,  and  tlie  same  as  used  unfolding  the  central  section  at  Fig.  6 — the 
joints,  too,  are  all  alike.  Fig.  8  is  the  concave  side-mould,  S  V  Z  the  six  parts  taken  from  Fig.  7,  and  Fig.  9 
is  the  convex  side-mould,  2,  6,  9,  the  six  parts  taken  from  2,  6,  9  of  Fig.  7.    See  Plate  46. 


Plate  No.  77 


Plate  No.  78 


Scale 


1^    I  N  =  1  Ft. 


PLATE  78. 

It  is  desirable  with  this  plan  of  stairs  to  get  the  wreath-piece  out  in  one  piece.  To  do  this  it  becomes 
necessary  to  force  the  joints  and  change  curves  to  tangent  the  inclination  of  the  straight  hand-rail.  Through 
Figs.  3  and  4,  Plate  31,  this  plan  is  treated  in  two  pieces  without  tlie  necessity  of  forcing  joints  or  curves  ; 
but  it  takes  more  time  to  work  two  twists  of  this  character  than  it  does  one  ;  besides,  the  two-piece  wreath 
makes  a  pronounced  ogee-like  curve  that  is  unpleasant  to  the  eye.  The  fault  lies  in  the  plan  ;  the  risers  are 
not  in  a  position  to  produce  the  best  form  of  wreath.  See  Plate  37,  Tigs.  5,  6  and  7,  for  the  best  plan  and 
wreath-piece. 

Fig.  I.  Plan  of  Quarter  Platform  Stairs  with  Quarter  Cylinder  8"  Radius,  and  the  Risers 
in  Connection  with  the  Platform  Placed  at  the  Chord  Lines. — M  A  V,  N  C  W  is  the  plan  and  centre 
line  of  rail  ;  A  B  C  is  the  plan  tangents  ;  prolong  A  B  to  P  indefinitely  and  C  B  to  J  indefinitely;  connect 
B  F,  the  governing  level  line  ;  divide  the  centre  line  A  C  into,  say,  six  equal  parts,  and  through  each  of  these 
draw  radial  lines  from  the  centre  F  ;  again  through  each  of  these  parts  draw  lines  parallel  to  F  B.  To  find 
the  heights  and  inclination  of  tangents  set  up  the  elevation  of  plan. 

Fig.  2.  Elevation  of  Treads  and  Rises  as  Given  at  Plan  Fig  i. — The  line  of  platform  ABC 
must  equal  the  plan  tangents  ABC,  Fig.  i.  At  X  X — the  centres  of  short  balusters  above  and  below  the 
platform— draw  the  bottom  line  of  rail  and  set  off  the  thickness  ;  also  draw  a  centre  line  at  D  and  I  ;  make 
D  A  and  S  I  each  2"  or  not  more  than  3"  ;  connect  D  I  ;  at  A  parallel  to  X  C  draw  the  line  A  M  ;  bisect  M  S  at 
N  ;  make  N  P  parallel  to  M  L  ;  from  B  draw  B  P  parallel  to  C  N  ;  through  D  draw  E  G  at  right  angles  to  P  A  ; 
and  from  E  draw  E  F  at  right  angles  to  X  X  ;  through  I  draw  H  J  at  right  angles  to  S  P  ;  from  H  draw  H  K  at 
right  angles  to  X  X  ;  at  Fig.  i  make  CDS  equal  M  N  S  of  Fig.  2,  and  draw  U  P  parallel  to  C  B  ;  connect 
P  S  ;  make  B  J  equal  B  P  ;  connect  J  A  ;  parallel  to  B  J  draw  1 ,  6  and  0  K  ;  ])arallel  to  C  T  draw  G  R  and 
H  Q.  To  find  the  angle  that  will  square  the  ivreath-picce  at  both  joints: — On  U,  as  centre,  describe  the  arc  R  T 
tangent  to  S  P  ;  connect  T  P  ;  then  the  bevel  at  T  indicates  the  angle  sought  ;  from  A  through  C  draw  A  E 
indefinitely  ;  with  P  S  as  radius,  on  B  as  centre,  describe  an  arc  at  E. 

Fig-  3.  To  Draw  the  Face-mould  -.—Make  D  E,  D  E  each  equal  D  E  of  Fig.  i  ;  at  right  angles  to 
D  E  draw  D  P  ;  make  D  P  equal  D  B  of  Fig.  i  ;  connect  P  E,  P  E  ;  make  E  Q,  E  Q  each  equal  D  A  or  S  I  ©f 
Fig.  2  ;  make  the  joints  at  Q,  Q  at  riglit  angles  to  P  Q  ;  make  G  L  N  equal  G  L  N  of  Fig.  i  ;  make  H  8,  9 
equal  the  same  at  Fig.  i  ;  make  P  Y  D  equal  B  Y  D,  Fig.  i,etc.  ;  through  E  draw  N  R  ;  make  E  R  equal  E  N  ; 
trace  the  curved  edges  through  the  points  thus  found  to  complete  the  face-mould.  The  angle  for  squaring 
the  wreath-piece  at  both  joints  is  found  at  T,  Fig.  i.  A  face-mould  of  this  character  is  given  in  full  detail 
at  Plate  ii. 

Figs.  4  and  5.  Convex  and  Concave  Side-moulds.— Fig.  4  is  the  concave  ;  M  N  is  the  six  parts 
taken  from  M  N,  Fig.  i.  The  heights  are  taken,  as  the  corresponding  letters  indicate,  at  Fig.  i.  The  joints 
are  the  same  as  shown  and  lettered  alike  at  Fig.  2.  Fig.  5  is  the  convex  :  VW  is  the  six  divisions  taken  from 
VW  of  Fig.  i.  The  heights  are  the  same.  The  joints,  as  indicated  by  the  corresponding  letters,  are  alike. 
The  departure  from  tlie  exact  unfoldment,  as  seen  by  the  circles,  is  made  necessary  because  the  joints  and 
curves  are  forced.  When  the  wreath-piece  is  squared  up  by  the  use  of  angle  T,  Fig.  i,  and  the  side-moulds 
to  insure  its  correct  curves,  then  the  material  F  G  and  J  K  is  cut  away,  leaving  the  joints  E  F  and  K  H  that 
connect  properly  with  the  straight  rail. 


Note.— The  thickness  of  plank  required  to  work  out  a  wreath-piece  is  invariably  given  by  the  squaring  shown  at  the 
ends  laid  out  from  the  centre  ;  and  the  unfoldment  of  its  natural  geometrical  curves  are  from  the  centre,  and  these  curves  are 
contained  within  the  limits  of  the  two  planes  of  a  plank  of  which  the  two  joints  of  a  wreath-piece  are  only  end  sections.  But 
in  any  case  of  this  kind  where  the  form  of  a  wreath-piece  is  forced  out  of  its  natural  curve-line  of  geometrical  unfoldment,  it 
takes  more  thickness,  how  much  more  will  be  determined  by  whatever,  at  both  ends,  or  either  end,  the  forced  form  of  side 
mould  is  outside  of  and  beyond  that  of  the  natural  geometrical  unfolded  curves. 


PLATE  79. 

At  Plates  34  and  35  two  methods  of  treatment  are  given — one  for  a  twelve-inch  and  the  other  for  a 
seven-inch  cylinder — both  for  the  toj)  of  straight  flights  of  stairs  ;  the  wreaths  to  be  worked  in  one  piece, 
making  a  complete  semicircle.  The  case  here  jjresented  is  also  to  be  worked  in  one  piece.  All  of  this  kind 
of  work  has  to  be  more  or  less  forced.*  What  I  mean  by  forced — as  applied  to  hand-railing — is  to  be  seen  in 
all  of  these  cases  by  the  changed  form  required  from  the  natural  geometrically  unfolded  centre  line  of  wreath 
when  put  in  comparison,  as  at  Figs.  3  and  4  of  this  Plate.  I  suggest  that  the  skilled,  intelligent  mechanic 
would  do  well,  as  a  rule,  not  to  decide  hurriedly  with  a  snap  judgment  that  this  or  that  kind  of  proceeding  or 
method  when  brought  to  his  notice  is  "no  good"  or  "  unprofitable."  It  would  be  better  to  give  a  little  of  his 
spare  time  in  careful  examination,  and  in  this  case  of  hand-railing,  for  instance,  work  a  wreath  up  as  explained 
out  of  some  soft  wood,  and  so  get  positive  knowledge  one  way  or  the  other. 

Fig.  I.  Plan  of  Half-turn  Platform  Stairs. — The  treads,  10";  rise,  8";  cylinder.  7";  hand-rail, 
4"  X  2!"  ;  the  wreath  to  be  worked  in  one  piece  ;  at  right  angles  to  3,  3  draw  3J,  DF,  4K,  A8,  4H,  BG  and 
3  L  indefinitely  ;  divide  the  centre  line  D  W  B  into  six  equal  parts,  and  through  these  draw  radial  lines  from 
the  centre  A. 

Fig.  2.  Elevation  of  Rises  and  Treads  from  Plan  Fig.  i. —  Draw  the  bottom  line  of  rail 
above  and  below  the  platform  at  X  X,  X  X,  the  centres  of  short  balusters  ;  set  off  D  C,  D  C  the  centre  of  rail, 
also  at  E  and  A  above,  the  thickness  of  rail  parallel  to  X  X  ;  through  E  and  E  draw  A  A  parallel  to  line  of  rise  ; 
parallel  to  tread  lines  draw  the  following :  A  B  K,  D  F,  E  J ,  E  L,  D  G  and  BAH;  at  Fig.  i  connect  F  G  ;  make 
F  K  and  F  J,  G  L  and  G  H  each  equal  half  the  thickness  of  the  wreath-plank  ;  draw  the  lines  H  J  and  L  K  ; 
make  8,  7  equal  0  D  of  Fig.  2  ;  make  all  the  measurements  for  face-mould  from  the  line  3  A  3  and  set  them 
from  line  5,  6  ;  as,  for  example,  make  7  YZ  equal  A  2,  2,  etc.  ;  through  the  points  so  found  trace  the  curved 
edges  of  the  face-mould  ;  make  C  C,  Fig.  i,  eijual  C  C  at  the  elevation.  Fig.  2  ;  connect  C  Q,  and  parallel  to 
C  C  through  the  points  of  divisions  on  centre  line  D  W  B  draw  lines  touching  C  Q. 

Figs.  3  and  4.  Convex  and  Concave  Side-moulds. — Fig.  3  is  the  convex-mould  ;  C  Q  is  the  six 
parts  taken  from  3,  2,  3,  Fig.  i  ;  the  heights  C  C,  0  M,  V  P,  etc.,  are  also  taken,  as  indicated  by  corresponding 
letters  at  Fig.  i.  The  joints  are  the  same  as  at  Fig.  2.  The  broken,  or  dotted,  lines  are  the  natural  unfolded 
geometrical  curve  lines,  from  which  a  forced  departure  has  to  be  made  as  shown.  Fig.  4,  the  concave  side- 
mould  ;  4,  2,  4,  is  the  six  parts  taken  from  4,  2,  4,  Fig.  i  ;  the  heiglits  are  the  same  as  those  of  the  convex 
side-mould  ;  the  joints  are  the  same  as  at  Fig.  i  and  Fig.  3.  This  wreath-piece  is  cut  square  through,  as 
shown  by  the  line  shading  at  joints.  The  joints  G  and  F  are  made  scjuare  from  tlie  face  of  the  plank,  centred 
and  lined  out,  as  seen  in  connection  with  Figs,  i  and  2.  At  the  joints  G  and  F  the  lower  level  squaring  line 
H,  and  the  upper  level  squaring  line  K,  the  slabs  should  be  cut  off  square  to  the  joints.  Then  resting  alternately, 
these  cut  surfaces  on  a  saw-table,  the  sides  may  be  cut  plumb  with  a  band-saw,  and  when  cleaned  up  apj)ly 
and  mark  the  lines  given  by  the  side-moulds.  To  cut  the  joints,  set  a  gauge  equal  to  A  B,  Fig.  2,  and  run  it 
against  the  squared  joints  G  and  F,  gauging  on  the  level  at  K,  also  at  H.  Next  cut  away  the  material  from 
gauged  lines  to  line  J  and  to  line  L,  making  the  joints  E  B,  E  B  at  Fig.  2.  The  top  and  bottom  of  the  wreath 
may  now  be  squared  with  the  band-saw  by  rolling  it  on  the  saw-table  on  its  convex  plumbed  side. 


Noi  K. — The  thickness  of  plank  required  to  work  out  a  wreath-piece  is  invariably  given  by  the  squaring  shown  at  the  ends 
or  butt-joints  as  laid  out  from  the  centre,  and  the  unfoklment  of  a  wreath's  natural  geometrical  curves  are  from  the  centre  ; 
and  these  curves  are  therefore  contained  within  the  limits  of  the  two  planes  of  a  plank,  of  which  the  joints  of  a  wreath-piece 
are  but  end  sections.  In  any  case  where  the  form  of  wreath  is  foiced  out  of  its  natural  geometrical  unfoldment  it  lakes  more 
thickness;  how  much  inore  will  be  determined  by  whatever  distance  at  both  ends  or  either  end  the  forced  form  of  side-mould 
is  outside  of  the  natural  geometrically  unfolded  curves.  Set  the  last  mentioned  distance  from  a  point  touching  the  lower  face 
of  plank,  down  the  plumb  line  J  or  H,  Fig.  2,  and  measure  for  half  the  thickness  required  at  right  angles  to  the  line  G  F. 


Plate  No. 79 


Scale  1>&]n  =  I  Ft 


Plate  No.  80 


S  C  ALE  1    I  N  -1  Ft. 


PLATE  80 


Circular  Stone  Work. 

Fig.  I.    Plan  of  Stone  Work  a  Quarter-circle,  P  C  F,  P  D  G,  with  Projecting  Capstone.— 

E  A,  the  thickness  of  stone  wall  ;  F  C,  the  width  of  cajjstone.  The  top  of  the  stone  wall  with  capstone  is 
recpiired  to  start  from  a  level  and  finish  to  a  level,  rising  the  height  J  L,  with  geometrically  reversed  curva- 
tures ;  all  within  the  quarter-circle  plan.  To  prepare  this  plan  for  unfolding  the  convex  and  concave  faces 
of  the  stone  and  its  top  curvatures  : — Divide  the  centre  line  I  M  Q  into  six  equal  parts  ;  throught  each  of 
these  divisions  draw  lines  from  the  centre  P  ;  from  M  at  right  angles  to  M  P  draw  J  K  ;  at  right  angles  to 
P  Q  draw  Q  K  ;  at  right  angles  to  P  E  draw  I  J  ;  parallel  to  P  M  N  draw  J  L  ;  make  J  L  any  required  height  ; 
connect  L  K  ;  parallel  to  I  J  draw  R  0  and  S  7  ;  from  X  and  from  V  parallel  to  Q  K  draw  V  8  and  X  W  • 
parallel  to  J  L  draw  0  T,  7  U,  8  Z  and  W  9. 

Figs.  2  and  3.  Unfoldment  and  Elevation  of  the  Convex  and  Concave  Faces  of  Stone 
Wall  and  Capstone  on  the  Lines  E  H  and  A  B,  Fig.  i.— At  Fig.  2  mark  on  the  line  J  K  the  six  parts 
of  the  concave  line  A  B,  Fig.  i  ;  from  each  of  these  set  up  perpendicular  heights  J  L,  0  T,  7  U,  etc.  ;  taken 
from  the  corresponding  letters  at  Fig.  i  from  the  points  LT,  U  N,  etc.,  describe  circles  equal  in  diameter  to 
the  thickness  of  capstone  ;  trace  lines  touching  the  circles  for  the  top  and  bottom  curves  of  capstone  ;  the 
centre  joint  at  N  is  made  at  right  angles  to  the  angle  L  of  Fig.  i — copy  the  angle  L,  as  at  N  V  V,  Fig.  2  ;  fix 
the  number,  size  and  joints  of  stone  as  required.  Fig.  3  is  the  unfolded  convex  face  of  stone  and 
curvatures  of  capstone  on  the  line  E  H,  Fig.  i.  The  six  parts  on  the  line  J  K  of  this  figure  are 
taken  from  the  six  divisions  at  E  H,  Fig.  i  ;  also  the  heights  are  the  same  as  indicated  by  the  corresponding 
letters.  From  the  fixed  joints  PEG,  Fig.  2,  draw  horizontal  lines  touching  F  E  G  of  Fig.  3.  Fig.  4. — Plan 
of  stone  wall  P  A  E,  P  B  H  from  Fig.  i  introduced  for  the  purpose  of  showing  presently  in  plan  the  exact 
sizes  of  each  of  the  four  stones  next  to  capstone,  as  follows  : — Take  stone  I,  Fig.  2  ;  apply  X  2  to  X  2, 
Fig.  4,  and  from  centre  P  draw  2,  2  ;  then  the  convex  2,  0  is  X  2,  stone  I,  Fig.  3  ;  connect  2  F,  the  joint- 
then  X  0,  2,  2  of  Fig.  4  is  the  size  in  plan  of  stone  I.  To  find  the  size  in  plan  of  stone  II:— At  Fig.  3 
drop  the  perpendicular  F  Z  and  prolong  4,  2  to  Z  ;  apply  2,  4  to  2,  4,  Fig.  4  and  draw  P  4,  4  ;  draw  P  Z  ; 
then  Z4,  4  is  the  size  of  stone  I  I.  To  find  the  size  of  stotte  I  I  I  : — Apply  4,6  of  Fig.  3  to  4,  6  at  the 
concave  Fig.  4  ;  draw  P  6  6  ;  make  4,  6  of  Fig.  3  equal  4,  6  of  convex  Fig.  4  ;  let  fall  the  perpendiculars 
G  A  and  E  B  ;  prolong  Y  6  to  A  and  6,  4  to  B  ;  make  6  A  and  4  B  of  Fig.  4  equal  the  same  of  Fig.  3  ;  draw 
P  A  and  P  B  ;  then  B  6,  6  is  the  size  of  stone  Mi;  make  6  Y  W  of  Fig.  3  equal  the  same  of  Fig.  4  ;  then 
D  AWC  is  the  size  of  stone  I  V.  7  he  drawing  is  now  prepared  to  make  the  concave  and  convex  templet  for 
the  four  stone.  The  unfoldment  of  the  curvatures  of  the  capstone  is,  as  before  stated,  on  the  face  lines  of 
the  stone  wall  A  B  and  E  H  of  Fig.  i,  and  the  only  use  of  it  here  is  to  get  the  exact  shape  of  the  wall  at 
X  F  EG,  etc.  ;  but  for  the  sides  of  the  capstone  as  side-moulds  for  which  it  is  required  it  must  be  unfolded 
on  the  lines  C  D  and  F  G,  Fig.  i.  Therefore,  that  portion  of  the  concave  and  convex  in  each  marked  M  K 
must  be  redrawn,  using  the  same  heights  and  joint  at  N,  l)ut  taking  three  of  the  divisions  from  the  line  C  D, 
Fig.  I,  for  the  concave  side-mould,  and  three  of  the  divisions  from  the  line  F  G  for  the  convex  side-mould. 

Fig.  5.  Plan  of  Capstone  taken  from  Fig.  i,  as  Shown  by  Corresponding  Letters,  to  be 
Prepared  for  Drawing  the  Face-mould.  — P  M  bisects  the  quarter-circle  I  Q  ;  from  M  draw  SV  at  right 
angles  to  M  P  ;  draw  I  S  at  right  angles  to  P  F  ;  from  S  draw  S  N  parallel  to  K  P  ;  make  S  N  equal  M  N  of 
Fig.  I  ;  connect  N  M  ;  prolong  S  I  to  B  and  to  K  indefinitely  ;  make  I  B  equal  S  N  ;  from  M  parallel  to  S  B 
draw  M  T  ;  connect  B  T  and  prolong  T  B  fo  A  indefinitely  ;  from  0  parallel  to  S  B  draw  0  E  and  W  LA  ; 
make  M  V  equal  S  R  ;  connect  V  K  ;  at  right  angles  to  S  I  draw  M  U  indefinitely  ;  on  S  as  centre  with  M  N 
as  radius  describe  an  arc  at  U  ;  connect  I  U. 

Fig.  6.  Face-mould  for  Capstone.— Make  I  U  etiual  I  U  of  Fig.  5  ;  make  I  Y  equal  S  I  of  Fig.  5  ; 
make  U  Y  equal  M  N  of  Fig.  5  ;  make  I  Z  equal  the  straight  from  L  to  the  joint,  or  from  K  to  joint  Fig.  3  ; 
through  I  draw  V  W  at  right  angles  to  Z  Y  ;  through  Z  parallel  to  P  W  draw  B  A  ;  make  I  K  equal  B  A  of  Fig 
5  ;  make  I  R  P  V  equal  B  E  H  T  of  Fig.  5  ;  draw  V  S,  R  T  and  K  D  parallel  to  Z  Y  ;  make  V  S  equal  T  J  of 
Fig.  s  ;  make  R  T,  I  C,  K  D  equal  C  0,  I,  5  and  L  W  of  Fig.  5  ;  through  U  draw  T  X  ;  make  U  X  equal  U  T  ; 
make  I  W  equal  I  P  ;  make  joint  U  at  right  angles  to  U  Y  ;  trace  the  curve  X  S  P  and  T  C  D  W,  to  complete  the 
face-mould.  The  angle  for  squaring  the  capstone  at  joint  Z  is  found  at  B,  Fig.  5,  and  the  angle  for  squaring 
at  joint  U  is  at  V  of  Fig.  5.  To  plumb  the  sides  of  the  capstone  the  face-mould  is  moved  on  the  face  of  the 
stone  along  the  line  B  A,  as  shown  by  the  dotted  lines.  After  the  sides  E  E,  F  F  of  the  capstone  piece  is  cut 
away  plumb,  then  the  side-moulds — made  as  explained  under  the  head  of  Figs.  2  and  t, — are  applied  ;  the 
concave  side-mould  to  the  side  E  E  and  the  convex  side-mould  to  the  side  F  F,  marking  from  these  the  top 
and  bottom  of  capstone  ;  thus  carrying  the  lines  of  squaring  through  from  joint  to  joint.  Two  of  these  pieces 
reversed  complete  the  capstone.  A  face-mould  of  this  character  is  given  in  full  detail  at  Plate  13.  For  a 
better  knowledge  of  side-moulds  study  Plates  20,  21  and  3. 


Note. — The  treatment  of  the  above  capstone  will  be  precisely  the  same  for  a  hand-rail,  if  required,  over  this  or  a 
similar  plan. 


PLATE  81. 


Plan  of  Circular  Stone  Staircase. 

The  shaded  space  around  the  wall  indicates  the  face-line  of  the  stone  wainscot  and  the  finished  plaster 
line.  The  outside  circular  line  is  the  face  of  brick  wall.  Into  this  brick  wall  the  stone  steps  are  bedded  eight 
inches,  and  resting  step  upon  step  lengthwise  on  the  horizontal  checks,  interlocking  each  other  with  warped 
joints  accurately  fitted  through  the  depth  of  each  stone,  form  an  archlike  self-sup])orting  structure.  The 
front  ends  of  steps  finisli  in  line  with  a  circle  five  feet  in  diameter  ;  the  nosings  projecting  on  the  line  L,  H,  N, 
10,  which  is  also  the  concave  line  of  the  stone  hand-rail.  The  hand-rail  is  seven  inches  wide  by  four  inches 
thick.  The  shaded  squares  show  the  position  and  size  of  balusters.  The  centre  line  of  hand-rail  S  R  U  TV 
passes  through  the  centres  of  balusters.  The  hand-rail  is  divided  into  four  pieces* — on  the  line  G  F,  the 
beginning  of  the  quicker  curve,  to  newel  S,  of  which  8  is  the  centre,  and  from  R  to  U  five  treads  are  included, 
for  the  joints  must  be  made  to  occur  at  the  centres  of  balusters  in  stone-work  ;  from  U  to  T  five  more  treads 
are  taken  in  the  third  piece  of  rail  ;  the  balance  from  T  to  V  makes  the  fourth  piece  ;  next,  the  tangents  have 
to  be  placed  and  the  heights  to  be  raised  over  each  tangent,  fixing  their  inclinations  ;  beginning  at  S,  draw 
S  E  at  right  angles  to  8  S  ;  at  R  draw  the  tangent  E  Q  at  right  angles  to  F  G  and  at  U  draw  the  tangent  Q  W 
at  right  angles  to  G  A  ;  at  T,  at  right  angles  to  G  I,  draw  W  B  ;  from  V,  the  centre  of  the  rail,  at  right  angles  to 
G  V  "draw  V  B  ;  to  fix  the  heights,  at  right  angles  to  R  Q  draw  Q  D  indefinitely  ;  at  right  angles  to  U  W 
draw  W  X  indefinitely  ;  at  right  angles  to  W  B  draw  B  C  indefinitely  ;  make  Q  D  two  and  a  half  rises  ;  con- 
nect D  R  ;  from  E  parallel  to  D  R  draw  E  F  ;  make  U  A,  W  X  and  T  I  each  two  and  a  half  rises  in  height  ;  con- 
nect A  Q,  X  U  and  I  W  ;  prolong  W  I  to  C  ;  from  I  parallel  to  T  B  draw  1 ,  7. 

Fig.  I.  Template  from  Plan  of  Circular  Band  to  be  Applied  on  the  Soffit.— An  example  of 
a  portion  of  th&  middle  band  will  show  how  to  ])roceed  for  either  of  the  circular  bands.  From  the  centre  of 
the  plan  G  describe  the  centre  line  of  the  middle  band  A  C  ;  then  draw  a  straight  line  from  A  to  C  ;  at  right 
angles  to  A  C  draw  A  B  ;  make  A  B  equal  three  rises  ;  connect  B  C  ;  anywhere  along  the  line  A  C  draw  i)er- 
pendiculars  X  P,  Q  R  and  V  S  ;  at  right  angles  to  B  C  draw  P  F,  R  G  and  S  T  ;  make  S  T  R  G  and  P  F  ecjual 
V  E,  Q  Z  and  X  0  ;  on  the  points  B  F  G  T  C  as  centres  describe  circles  in  diameter  equal  to  the  width  of  band, 
and  touching  these  circles  trace  the  curved  edges  of  band. 

The  hand-rail,  the  wainscoting,  the  heavy  base  and  cap  mouldings,  the  unfolding  and  paneling  of  the 
soffit  and  the  curves,  angles  and  joinings  of  the  stone  steps  have  each  to  be  treated  specially  in  one  of  the 
four  more  Plates  following  this  one. 

See  Plate  7,  Fig.  10  ;  also  Plates  53  and  54. 


*The  divisions  of  hand-rail,  if  thought  desirable  for  any  reason,  may  be 
each,  leaving  the  top  piece  of  rail  to  take  the  odd  tread. 


; — Commencing  at  R  three  pieces  of  three  treads 


Plate  No. 81 


Plate  No. 82 


PLATE  82. 


CIRCULAR    STONE  STAIRS. 

Elevation  of  Sections  of  Steps.  Their  Connections,  Curves,  and  Joints  from  which  to  Make 
Templets.  Also  Sections  and  Elevation  of  Wainscot  ;  Its  Construction,  together  with 
Management  of  Base  and  Cap  Mouldings. 

Figs,  i  and  2. — Elevation  of  sections  of  the  first  five  steps.  Fig.  2  is  taken  at  the  face  of  wainscoting, 
Z  M  K  0,  Plate  81.  Fig.  i  is  taken  at  the  line  of  nosing,  L,  H,  N,  1  0,  Plate  81.  The  check  l^"  A  B,  Fig.  i, 
is  set  from  rise  line  to  H  at  plan,  Plate  81,  and  the  dotted  line  G  H  J  drawn  to  get  the  radial  distance  at  K  J, 
the  face  of  wainscot,  which  is  3+".  This  may  be  diminished  to  2V ,  which  is  done  at  C  D,  Fig.  2,  as  it  seems 
an  unnecessary  amount  of  check,  and  some  little  stone  is  saved,  too.  The  joint  will  not  radiate  truly,  but  if 
it  does  not  vary  from  the  horizontal,  as  it  will  not  when  treated  as  directed,  it  will  make  no  perceptible  differ- 
ence. Through  the  angle  of  tread  and  riser  F  and  A,  Fig.  i,  draw  the  line  FA  ;  from  B  at  right  angles  to 
F  A  draw  a  line  the  required  depth  of  the  face  end  of  the  stone  step  B  to  E  ;  through  E  parallel  to  F  A  draw 
G  R  ;  on  this  face  all  the  joints — except  joints  through  curves — are  parallel  to  B  E,  or  at  right  angles 
to  F  A  ;  at  Fig.  2  draw  the  line  XC  ;  at  right  angles  to  X  C  from  D  draw  D  P  indefinitely;  from  E, 
Fig.  I,  draw  the  horizontal  line  E  P  ;  then  P,  Fig.  2,  touching  the  joint  D  P  fixes  the  depth  of  stone  at  S  P 
from  V  E  of  Fig.  i  ;  from  P  at  Fig.  2  parallel  to  X  C  draw  0  Q  ;  at  Fig.  i  prolong  the  floor  line  W  J  to  H  ; 
prolong  the  joint  at  G  to  H  ;  on  H  as  centre  describe  the  curve  G  J  ;  then  to  find  the  radial  from  J  to  the 
face  of  wainscot  section  Fig.  2,  take  W  J,  Fig.  i,  in  two  or  more  parts  and  set  them  at  Plate  81  ;  from  9  to 
L  and  from  G  through  L  draw  G  M  ;  then  P  M  will  be  the  radial  distance  to  set  in  front  of  the  fourth  riser 
X  W  at  Fig.  2,  Plate  82,  and  from  W  perpendicular  to  the  floor  line  draw  W  L  ;  then  produce  a  curve  0  L, 
finishing  to  the  floor  line  and  tangent  to  0  P  ;  the  horizontal  lines  G  N  and  K  M  place  the  joints  at  the  curve 
L  0.  Figs.  3  and  4. — Elevation  of  several  steps  from  the  top  of  the  flight  down,  which  show  in  the  checks, 
the  regular  joints  and  curve  joints  substantially  what  has  been  before  explained.  At  Fig.  3,  at  the  top  joint, 
it  will  be  preferable  to  cut  a  straight  joint  A  B  in  the  face  of  stone  about  three  inches  before  commencing  to 
cut  the  angular  joint. 

Fig.  5.  Cross  Section  of  Stone  Wainscot  wit  h  Base  G,  Base  Moulding  A  and  Cap  Mould- 
ing B. — The  line  of  finished  plaster  is  at  C,  the  face  of  brick-work  at  D.  This  section  is  given  on  the  line 
E  F,Fig.  6. 

Fig.  6.  Elevation  of  Wainscot. — G  G  G,  etc.,  is  the  joints  of  base,  the  face  of  which  is  worked  to 
the  circle  and  set  25"  in  front  of  the  face  of  wainscot,  as  at  G,  Fig.  5.  The  wainscot  may  be  got  out  and 
joined  as  at  H,  H,  H,  H.  A  cross-section  of  this  piece  of  panel-work  on  the  circle  is  shown  at  J  J  ;  I  is  an 
inserted  raised  panel  ;  5  is  the  centre  from  which  the  lower  portion  of  the  wainscot  panel  is  described,  extend- 
ing within  the  radial  lines  5  V  and  5  S.  The  cap-mouldings  T  M  0  and  the  base-mouldings  S  V  X  will  have 
to  be  squared  up  similar  to  hand-railing,  and  on  this  drawing  it  is  most  convenient  to  fix  lengths  of  pieces — 
material  considered — fit  to  handle  that  will  cover  in  number  the  total  length  required  ;  also  heights,  etc.,  must 
be  placed.  At  the  ca])-moulding  easement  draw  a  horizontal  line  from  the  centre  T  to  K  and  from  M  to  N  ; 
draw  the  perpendiculars  M  K  and  ON.  It  is  best  to  make  M  N  a  length  on  plan  of  which  any  certain  number 
of  pieces  will  complete  the  circle.  Divide  K  L  into  three  parts,  and  from  two  of  the  points  of  division  raise 
perpendiculars.  At  S,  the  centre  of  the  base-moulding,  draw  the  horizontal  line  S  P,  and  from  V  draw  the 
horizontal  line  V  W  ;  draw  the  perpendiculars  V  P  and  X  W  ;  divide  P  R  into  four  parts,  and  from  three  points 
of  the  division  raise  perpendiculars.    Fig.  7. — Moulded  lower  edge  at  finished  front  end  of  steps. 


*  Sometimes  two  centres  have  to  be  used — one  of  less  radius  for  base  moulding  and  lower  part  of  first  and  second  panels 
and  base  moulding — and  then  these  panels  are  possibly  of  a  greater  height  than  the  others. 


PLATE  83. 


Unfolded  Soffit  of  a  Circular  Stone  Staircase. 


Unfolding  the  Soffit  of  a  Circular  Stone  Staircase,  and  Apportioning  the  Panels,  etc. — 

Draw  a  line  A  B  indefinitely  ;  the  centre  E  will  not  be  needed  in  this  development  until  later  on  ;  make  A  B 
equal  I,  0  of  the  \Aan  at  Plate  8i  ;  make  B  C  equal  P  Q  of  Fig.  2,  Plate  82  ;  make  A  D  equal  E  R,  Fig.  i, 
Plate  82  ;  at  the  plan  Plate  81  draw  N  0,  and  at  right  angles  to  N  0  draw  the  perpendicular  0  F  ;  connect 
F  N  ;  take  F  N  for  radius,  and  on  A — of  this  Plate — as  centre  describe  an  arc  at  C,  and  on  B  as  centre  describe 
an  arc  at  D  ;  at  the  intersection  of  the  arcs  at  C  and  D  draw  the  line  D  C  ;  proceed  in  like  manner  from  D 
and  C,  as  before,  from  A  and  B  for  the  number  of  six  steps  ;  then  draw  the  line  G  F  ;  again  from  A  B  proceed 
and  describe  intersecting  arcs  at  fK  and  L,  and  so  on  for  five  steps  to  M  and  N  ;  now  the  centre  may  be  found 
by  drawing  the  line  M  N  until  it  intersects  the  line  G  F,  prolonged  to  meet  at  the  centre  E  ;  with  E  A  as  radius 
describe  the  line  of  nosing  J  A  0  ;  with  E  B  as  radius  describe  the  face- line  of  wainscot  ;  with  E  W  as  radius 
describe  the  finished  plaster  line  ;  mark  at  F  J  the  two  steps  E  B,  Fig.  3,  Plate  82  ;  mark  also  at  G  H  the 
two  steps  G  D,  Fig.  4,  Plate  82  ;  connect  Q  R  and  J  H  ;  mark  at  N  0  the  three  steps  J  S,  Fig.  i,  Plate  82  ; 
mark  also  at  M  P  the  three  steps  L  0,  Fig.  2,  Plate  82  ;  connect  S  T,  U  V  and  OP;  A  Z  is  the  projection  of 
nosing  ;  Z  X  is  the  moulded  lower  edge  on  the  finished  face  of  the  front  ends  of  steps.  The  drawing  will 
furnish  its  own  explanations  by  measurements  and  observation  of  circular  and  radial  bands,  division  of  panels, 
etc.  It  is  not  pretended  that  this  unfoldnient  of  a  warped  surface,  i.e.,  the  soffit,  is  geometrically  exact,  but 
is  a  near  approach,  and  serves  a  useful  purpose  for  at  least  j^lanning  and  viewing  the  paneling  on  the  ap- 
proximately unfolded  surface. 


Plate  No.  84 


PLATE  84. 


Hand-railing  for  Circular  Stone  Staircase. 

Fig.  I. — Elevation  of  treads  and  rise  sufficient  in  number  to  show  the  height  of  hand-rail,  length,  and 
position  of  balusters,  and  placing  the  heights  A  B  and  C  D  at  the  top  and  bottom  of  the  flight,  giving  the 
position  of  D  and  A  above  the  floors.  The  joints  E  and  F  show  their  place  in  stone-work  to  be  at  the  centre 
of  baluster.  At  Fig.  io,  Plate  76,  drawings  and  explanations  are  given  to  cut  the  balusters  to  an  exact  length, 
the  tops  to  fit  the  7varped  bottom  of  rail.  Between  the  centres  of  these  two  balusters  there  are  five  treads,  as  at 
plan  Plate  81.  At  the  top  the  baluster  sets,  as  at  plan,  on  the  third  tread  down  ;  so  also  at  the  bottom  the 
baluster  sets,  as  at  plan,  on  the  fourth  tread  up.  The  treads  H  J  are  taken  on  the  centre  line  of  rail  3,  3, 
Plate  81.  From  F  and  from  E  draw  the  horizontal  lines  E  B  and  F  C  ;  CD  equals  7  C  and  B  A  equals  R  F, 
both  of  Plate  81.* 

Fig.  2.  Plan  of  First  Wreath-piece  8  S  R,  Plate  81,  to  be  Prepared  for  Drawing  the  Face- 
mould. — Make  H  C  at  right  angles  to  D  H  and  equal  A  B,  Fig.  i  ;  connec^t  C  D  ;  parallel  to  the  tangent  A  D 
draw  H  G,  L  N,  W  X  U,  0  P  and  S  R  T  ;  make  E  G  equal  H  C  ;  connect  G  A  ;  then  the  bevel  at  G  i^idicates 
the  angle  for  squaring  the  wreath-piece  at  joint  over  A.  The  angle  for  squaring  the  tvreath  over  joint  H  is  found 
as  folloivs : — Prolong  F  H  to  J  ;  jjrolong  A  D  to  J  ;  on  H  as  centre  with  H  K  as  radius  describe  the  arc  K  I  ; 
connect  I  J  ;  then  the  bevel  at  I  indicates  the  angle  sought. 

Fig.  3.  Face-mould  from  Plan  Fig.  2  ;  also  Showing  the  Squaring  of  the  Wreath-piece 
at  Both  Joints. — Draw  tlie  line  T  N  ;  make  A  R  equal  A  R,  Fig.  2  ;  make  A  P  U  N  equal  the  same  at  Fig. 
2  ;  at  right  angles  to  T  N  draw  R  S,  A  D,  U  W  and  N  L  ;  make  A  D  equal  A  D  of  Fig.  2  ;  also  A  B  equal  A  B 
of  Fig.  2  ;  make  D  B  equal  D  C  of  Fig.  2  ;  make  R  S  equal  T  S  of  Fig.  2  ;  make  N  Y  L  and  U  X  W  equal 
M  Y  L  and  V  X  W  of  Fig.  2  ;  through  B  draw  L  0  ;  make  B  0  equal  B  L ;  make  A  T  equal  A  P  ;  make  the 
joint  B  at  right  angles  to  D  B  ;  through  P  X  Y  0  and  T  S  D  W  L  trace  the  curved  edges  of  the  face-mould. 
Squaring  the  wreath-piece  at  joint  A  is  done  with  the  angle  G,  Fig.  2,  and  at  joint  B  with  the  angle  I,  Fig.  2. 
The  dotted  lines  show  the  movement  of  the  face-mould  along  joint  A  to  plumb  the  sides  of  the  wreath-piece 
from  joint  to  joint.  Side-moulds  are  given  for  this  wreath-piece  at  Plate  76,  Figs.  4,  5  and  6.  A  face-mould 
of  the  above  character  is  explained  in  detail  at  Plate  13.  See  also  Plate  32,  Figs.  4,  5  and  6  ;  also  Note 
appended.    Side-moulds  for  this  case  are  given  at  Plate  76,  Figs.  4,  5  and  6. 

Fig.  4.  Plan  of  Second  and  Third  Wreath-pieces  R  U  and  U  T  at  Plate  81  to  be  Prepared 
for  Drawing  the  Face-mould. — Make  the  perpendiculars  E  D  and  G  K  each  two  and  a  half  rises  ;  connect 
D  G  and  K  H  ;  through  E  draw  H  J  C  ;  on  G  as  centre  with  G  D  as  radius  describe  the  arc  D  C  ;  parallel  to 
G  A  draw  R  Z,  Q  Z  and  I  Z  ;  parallel  to  E  D  draw  X  P,  L  0  and  M  N  ;  prolong  G  E  to  B  ;  on  E  as  centre 
describe  the  arc  F  B  ;  connect  B  A  ;  then  the  bevel  at  B  indicates  the  angle  with  which  to  square  the  ivreath-piece 
at  both  Joints. 

Fig.  5.  Face-mould  from  Plan  Fig.  4,  also  Showing  the  Squaring  of  the  Wreath-piece  at 
Both  Joints. — Draw  the  line  N  C  ;  make  J  C  and  J  N  each  ecpial  J  C  of  Fig.  4  ;  make  J  M  at  right  angles 
to  N  J  and  equal  to  J  G  of  Fig.  4  ;  make  C  G  H  L  and  N  G  H  L  each  equal  D  P  0  F  N  of  Fig.  4  ;  at  right 
angles  to  N  C  draw  lines  through  each  of  these  points  ;  make  G  F,  G  0,  H  0,  H  X,  L  0  and  L  X  equal  X  Z, 
X  I,  L  0  and  L  X  equal  X  I,  X  Z,  L  Z,  L  Q,  M  Z  and  M  R  of  Fig.  4  ;  through  C  draw  F  0  ;  make  C  0  equal 
C  F;  trace  the  curved  edges  of  face-mould  through  the  points  thus  found,  and  make  the  joints  C  and  N 
at  right  angles  to  the  tangents. 

Fig.  6.  Squaring  the  Wreath-piece  with  Face-mould,  Fig.  5,  and  the  Angle  at  B,  Fig.  4. — 
The  dotted  lines  show  the  position  of  face-mould  in  squaring  the  wreath-piece  as  it  is  moved  on  the  slide- 
line  from  Z  to  X.  A  side-mould  for  this  wreath-piece  for  the  convex  and  concave  is  a  straight  parallel  strip 
of  sheet  zinc  of  a  width  equal  to  the  thickness  of  rail  and  long  enough  to  reach  from  joint  to  joint  around  the 
convex.  The  face-mould,  Fig.  5,  is  explained  in  detail  at  Plate  15.  The  case  of  hand-rail,  Figs  2  and  3, 
limited  by  length  of  tangents,  cannot  always  be  made  to  answer  the  required  height  in  this  geometrically  easy 
way,  and  for  this  reason  we  are  obliged  to  use  other  and  less  desirable  methods,  as  follows  : 

Fig.  7.  Plan  of  Wreath-piece  same  as  Fig.  2  to  be  Prepared  for  Drawing  a  Face-mould  that 
will  Bring  the  Rail  to  a  Lower  Level,  as  Much  as  A  G,  Fig.  i.— Let  U  Y  B  equal  H  C  D,  Fig.  2  ;  let  B  E 
ecpial  A  G,  Fig.  i  ;  from  F  draw  F  R  parallel  to  B  U  ;  make  F  R  equal  B  U ;  make  Y  S  equal  B  E  ;  draw  S  K  parallel 
to  U  B  ;  parallel  to  U  Y  draw  K  J  ;  from  J  draw  J  R,  the  governing  level  line  common  to  both  planes  ;  parallel 
to  J  R  draw  V  0  6,  5  Q  L,  3,  2  N,  etc.  ;  at  right  angles  to  J  R  draw  U  W  and  FX  indefinitely  ;  on  B  as  centre, 
with  B  Y  as  radius,  describe  the  arc  Y  W  ;  and  again  on  8  as  centre,  with  E  F  as  radius,  describe  an  arc  at  X  ; 
connect  X  W  ;  parallel  to  J  R  draw  B  M  indefinitely  ;  prolong  Y  T  to  M  ;  prolong  B  U  to  H  ;  on  U  as  centre, 
with  U  G  as  radius,  describe  the  arc  G  H  ;  connect  H  M  ;  then  the  angle  at  H  zvill square  the  wrealh-piece  over 
joint  U.  The  angle  for  squaring  the  wreath-piece  over  joint  F  will  be  found  on  the  plumb  face  of  joint 
presently  to  be  explained.  Make  D  C  half  the  thickness  of  stone  used  to  get  out  the  wreath-piece,  and  from 
C  draw  C  D  at  right  angles  to  F  E.  D  E  is  to  be  added  to  lower  joint  of  face-mould  for  extra  stone  required 
to  admit  of  cutting  a  plumb  joint  through  the  centre  of  thickness  at  that  end. 

Fig.  8.  Face-mould  from  Plan  Fig.  7.— Draw  the  line  X  W  ;  make  X  Z  W  equal  X  Z  W  of  Fig.  7  ; 
make  W  B  equal  Y  B  of  Fig.  7  ;  make  X  B  equal  F  E  of  Fig.  7  ;  connect  B  Z  ;  make  X  R  equal  D  E  of  Fig.  7  ; 
make  the  joints  R  and  W  at  right  angles  to  the  tangent  ;  make  W  I  G  K  equal  Y  I  G  K  of  Fig.  7  ;  make  X  P  2 
equal  F  P  2  of  FiG.  7  :  make  I,  V,  I,  6  equal  0  V,  0  6  of  Fig.  7  ;  make  P  5,  P  L,  2  N,  2,  3  equal  Q  5,  Q  L,  2  N 
and  2,  3,  etc.,  of  Fig.  7  ;  trace  the  curved  edges  of  the  face-mould  through  the  points  thus  found.  The 
slide-line  is  at  right  angles  to  B  Z.    A  face-mould  of  this  character  is  given  in  detail  at  Plate  16. 

Fig.  9.  Squaring  the  Wreath-piece  by  Face-mould,  Fig.  8,  and  the  Angle  at  H,  Fig.  7. — 
The  slide-line  must  be  squared  through  to  the  other  face  of  stone.  The  angle  for  squaring  the  wreath-piece 
at  joint  0  is  found  at  H,  Fig.  7,  and  controls  the  sliding  of  face-mould,  which  is  moved  up  along  the  slide- 
line,  as  shown  by  the  dotted  lines,  until  the  tangent  on  the  face-mould  touches  X,  the  point  of  plumb  line,  A 
at  A  B  on  the  joint  ;  then  mark  the  lower  point  P  ;  the  same  thing  is  done  on  the  other  face  of  stone  ;  the 
face-mould  being  moved  again,  but  downward,  along  the  slide-line  until  the  point  0 — the  tangent  point — 
squares  from  the  joint  to  B.  Fig.  id  is  a  sketch  merely  introduced  to  show  that  the  joint  P  through  the 
stone  is  cut  plumb,  as  at  F  Y,  and  the  easement  to  a  level  is  forced  in  the  manner  shown.  B  E  and  E  Y  is  the 
two  heights  B  E  and  U  Y  of  Fig.  7.  The  tangent  point  P,  Fig.  9,  if  marked  at  both  faces  of  the  stone,  as  it 
should  be,  is  represented  by  P  P,  Fig.  9,  and  governs  the  squaring  at  this  joint.  Proper  side-moulds — that  is, 
to  suit  this  case — are  given  for  the  wreath-piece  Fig.  9,  at  Plate  76. 

*  The  face-mould  for  the  wreath-piece  joining  at  F  and  raising  the  height  C  D  'S  not  given  because  Fics.  2  and  3  cover  it 
the  only  difference  being  is  that  it  i^  a  little  shorter  piece— TV  at  Pi.atf,  Si.    See  Plate  56  for  sliding  face-moulds,  etc. 


PLATE  85. 

Squaring  Base  and  Cap  Mouldings  for  Wainscot  of  Circular  Stone  Staircase. 

Figs.  I  and  2.  Plan  of  Base  Moulding  Shown  at  A,  Fig.  5,  and  at  SV  and  V  X  of  Fig.  6, 
Plate  82,  to  be  Prepared  for  Drawing  Face-moulds. — G  0  is  tlic  radius  to  face  of  wainscot  from  G  0 
It  plan  Plate  81.  At  Fig.  i  on  the  face  line  of  wainscot  make  J  L  etjual  S  R,  Fig.  6,  Plate  82  ;  take  also 
the  four  divisions  R  P  and  set  them  from  L  to  0,  and  from  0  to  N  set  off  the  four  divisions  V  W  at  F"ig.  6, 
Plate  82  ;  through  N  draw  Y  G  ;  through  0  draw  C  G  ;  draw  the  tangent  0  F  at  right  angles  to  0  G  ;  make 
0  B  equal  P  V  of  Fig.  6,  Plate  82  ;  draw  N  S  at  right  angles  to  N  G  ;  through  0  draw  S  F  at  right  angles  to 
0  G  ;  draw  S  K  perpendicular  to  0  S  ;  make  S  K  and  N  Y  each  equal  one  half  of  W  X  at  Fig.  6,  Plate  82  ; 
connect  Y  S  and  K  0  ;  parallel  to  K  0  draw  B  F  ;  from  F  draw  F  J,  and  prolong  J  F  to  C  ;  on  0  as  centre 
describe  the  arc  D  E  ;  connect  E  C  ;  then  the  bevel  at  E  indicates  the  angle  that  is  required  to  square  the 
moulding  at  the  joint  over  0  ;  from  6,  0  and  M  parallel  to  F  J  draw  6,  8,  0  1,  M  Q,  and  5  R  ;  make  H  I  equal 
0  B  ;  connect  I  J  ;  f/ien  the  bevel  at  I  indicates  the  angle  that  mill  square  the  moulding  at  the  joint  over  J  ; 
from  0  at  right  angles  to  F  J  draw  OA;  on  F  as  centre  with  F  B  as  radius  describe  the  arc  B  A  ;  connect 
J  A.  At  Fig.  2  draw  the  radial  S  9  indefinitely  ;  then  S  9  is  the  governing  level  line  ;  through  N  draw  the  line 
OX;  on  S  as  centre  with  S  Y  as  radius  describe  the  arc  Y  X  ;  prolong  S  N  to  L  ;  on'  N  as  centre  with  N  Z 
as  radius  describe  the  arc  Z  L  ;  connect  L  G  ;  then  the  bevel  at  L  shoics  the  angle  required  to  square  this 
piece  of  mouliling  at  both  joints  ;  parallel  to  S  9  draw  U  V  and  C  C  ;  at  right  angles  to  S  N  draw  V  W  and  C  P. 

Fig.  3.  Side-mould  for  Fig.  4. — Draw  the  line  A  C  B  ;  divide  5,  6  of  Fig.  i  into  four  equal  parts  ; 
set  these  parts  from  B  to  C  ;  make  C  A  equal  5  X  of  Fig.  i  ;  make  B  D,  W  S,  Y  R,  and  X  Z  ecjual  P  V,  0  0, 

0  0,  0  0,  Fig.  6,  Plate  82  ;  with  these  points  as  centres  describe  circles  equal  in  diameter  to  the  height  of 
base  moulding  ;  touching  the  circles,  trace  the  curves  of  tlie  concave  side-mould. 

Fig.  4.  Face-mould  from  Plan,  Fig.  i  ;  also  Showing  the  Squaring  of  this  Piece  of  Mould- 
ing at  Both  Joints. — Draw  the  line  K  L  ;  make  J  S  at  right  angles  to  K  L  and  equal  to  J  F,  F"ig.  i  ;  make 
J  0  equal  J  A  of  Fig.  i  ;  make  S  0  equal  F  B;  connect  0  S  ;  make  K  J  R  Q  I  L  equal  K  J  R  Q  I,  3  of  Fig. 

1  ;  parallel  to  J  S  draw  L  Z,  1,0,  Q  M,  R  N,  K  E,  and  make  these  equal  N  6,  H  0,  Z  M,  X  5,  J  L  of  Fig.  i  ; 
connect  R  N  and  K  E  ;  make  the  joint  0  at  right  angles  to  0  S  ;  trace  the  curves  as  found.  2 he  angle  to 
square  this  piece  at  joint  S  is  found  at  I,  Fig.  i,  and  for  joint  0  at  E,  Fig.  i.  In  marking  the  stone  see  that 
there  is  enough  width  equal  to  X  X,  X  X,  or  a  little  more,  taking  notice  that  more  width  is  required  from  the 
tangent  one  way  than  the  other.    At  least  one  face  of  the  stone  must  be  perfectly  true  and  out  of  wind. 

Fig.  5.  Face-mould  from  Plan  Fig.  2  ;  also  Showing  the  Squaring  at  Both  Joints.— Draw 
the  line  A  B  ;  make  C  A  and  C  B  each  equal  T  X  of  Fig.  2  ;  at  right  angles  to  A  C  draw  D  E  ;  make  C  D,  C  E 
equal  T  S,  T  R  of  Fig.  i  ;  connect  D  B  and  D  A  ;  make  D  G  0  and  D  G  0  each  ecjual  S  W  P  of  Fig.  2  ;  par- 
allel to  E  D  draw  S  0,  S  0,  F  G,  F  G  ;  make  0  S,  0  S  each  ecpial  C  C  of  Fig.  i  ;  make  G  Fand  G  F,  C  D  and  C  E 
each  equal  V  U,  F  R,  and  T  S  of  F'ig.  i  ;  make  the  joints  B  and  A  at  right  angles  to  the  tangents  ;  trace  the 
curves  through  the  points  found.  The  angle  to  square  this  piece  of  moulding  at  both  joints  is  given  at  L,  Fig.  2. 
This  face-mould  is  the  same  as  that  given  for  hand-rail  at  Plate  84,  Figs.  4,  5,  and  6,  and  also  at  Pla'i  e  15. 

Figs.  6  and  7.  Cap-moulding  of  Wainscot  Shown  at  Section  B,  Fig.  5,  and  at  T  M  0  of 
Fig.  6,  Plate  82. — M  0  is  the  radius  to  face  of  wainscot  ;  J  0  on  face  of  wainscot  is  taken  in  the  four 
divisions  K  T  of  Fig.  6,  Plate  82  ;  also  0  X  is  taken  in  four  parts  from  M  N  of  the  last-named  figure  and 
plate  ;  the  two  heights  at  Fig.  7,  5  S  and  X  W,  are  found  at  N  0  of  Fig.  6,  Plate  82  ;  draw  0  E  at  right 
angles  to  0  M  ;  make  the  height  0  4  ecjual  K  M  of  Fig.  6,  Plate  82  ;  parallel  to  S  0  draw  4  E  ;  connect  E  J  ; 
at  right  angles  to  E  J  draw  J  B;  parallel  to  E  J  draw  OH,  D6,  and  I  A;  make  G  H  equal  0  4;  connect 
H  J  and  prolong  to  F  and  to  A  ;  then  the  bevel  at  H  indicates  the  angle  required  to  square  the  moulding  at  joint  over 
J  ;  prolong  J  E  to  X  ;  make  0  R  equal  0  K  ;  connect  R  X  ;  then  the  bevel  at  R  gives  the  angle  to  square  the 
moulding  at  joint  over  0  ;  at  right  angles  to  J  E  X  draw  0  C  ;  on  E  as  centre  with  E  4  as  radius  describe  the 
arc  4  C  ;  connect  C  J  ;  at  Fig.  7  draw  5,  9  to  M  indefinitely  ;  then  5,  9  is  the  governing  line  ;  through  X  draw 
OP;  on  5  as  centre  with  5  W  as  radius  draw  the  arc  W  P  ;  prolong  5  X  to  Y  ;  on  X  as  centre  with  X  1  as 
radius  describe  the  arc  I  Y  ;  connect  Y  M  ;  then  the  bevel  at  Y  indicates  the  angle  required  to  square  the  moulding 
at  both  joints  over  X  a/id  0  ;  parallel  to  5,  9  draw  3  U  and  Q  Z  ;  parallel  to  X  W  draw  U  V  and  Z  Z. 

Fig.  8.  Face-mould  from  Plan  of  Moulding  Fig.  6  ;  also  Showing  the  Squaring  at  Both 
Joints. — Draw  the  line  F  A  ;  make  F  J  H  6  A  equal  the  same  of  Fig.  6  ;  at  right  angles  to  F  A  draw  A  I,  6  D, 
H  4,  J  E,  and  F  S  ;  make  F  S,  J  E,  H  4,  6  D,  and  A  I  equal  J  E,  G  0,  L  D,  and  B  i  of  Fig.  6  ;  make  the  joint  at 
4  at  right  angles  to  4  E  ;  trace  the  curve  lines.  The  angle  to  square  this  piece  of  moulding  at  joint  J  is  found 
at  H,  Fig.  6  ;  the  angle  for  joint  4  is  taken  from  R,  FiG.  6. 

Fig.  9.  Side-mould  for  Fig.  8. — Draw  the  line  KT;  divide  the  concave  L  D  I,  F'ig.  6,  into  four 
parts  and  set  them  from  K  to  T  ;  draw  K  M,  X  X,  X  X  perpendicular  to  K  T  ;  make  K  M,  X  X,  X  X  equal  the 
same  at  Fig.  6,  Plate  82  ;  through  the  points  thus  found  describe  circles  equal  in  diameter  to  the  thickness 
of  rail,  and  trace  the  curved  lines  touching  the  circles  ;  make  the  angle  M  0  8  equal  the  angle  4,  0,  8,  Fig.  6, 
and  make  the  joint  at  right  angles  to  M  8. 

Fig.  10.  Face-mould  from  Plan  Fig.  7;  also  Squaring  this  Piece  of  Cap-moulding  at 
Both  Joints. — Draw  the  line  A  B  ;  make  D  B  and  D  A  each  equal  P  T  of  Fig.  7  ;  at  right  angles  to  B  A  draw 
C  D  G  ;  make  D  C  equal  T  5,  Fig.  7  ;  connect  B  C  and  A  C  ;  make  the  joints  B  and  A  at  right  angles  to  the 
tangents  B  C  and  A  C  ;  make  B  Z  E  and  A  Z  E  each  equal  W  Z  V,  Fig.  7  ;  parallel  to  C  G  draw  Z  Q,  E  F  ;  make 
Z  Q,  E  F,  D  G  equal  Z  Q,3  U,  and  T  N  of  Fig.  7  ;  trace  the  curves.  The  angle  to  square  this  piece  of  mould- 
ing at  both  joints  is  taken  from  Y  at  Fig.  7  Where  stone  is  used — such  as  onyx — that  has  pronounced  bed 
lines,  then  in  case  of  mouldings  or  hand-rail  worked  from  a  face-mould  like  Fig.  8,  the  face  of  stone  requires 
such  extreme  inclination  that  the  bed  lines  are  thrown  on  top,  or  obliquely,  in  that  direction  ;  and  when 
joined  to  a  piece  of  moulding  worked  from  F'lG.  10,  or,  a  similar  piece  of  hand-railing,  where  the  inclination 
of  the  face  of  stone  is,  as  in  this  latter  case,  slight,  then  it  becomes  evident  that  the  two — the  first  mentioned 
of  which  the  bed  lines  are  nearly  vertical  and  the  latter  nearly  horizontal — will  not  answer  joined  in  that 
position  at  alb  The  remedy  for  this  mismatching  is  to  square  up  such  a  piece  as  F"iG.  8  out  of  soft  pine,  and 
copy  it  with  patterns  or  templets  marked  on  two  parallel  ])lane  surfaces  as  the  twisted  piece  lies  flat.  These 
parallel  plane  surfaces  toucli  the  twist  at  one  or  more  points,  and  the  space  between  the  flat  lying  piece  and 
the  planes  is  the  exact  thickness  of  stone  required,  and  no  less  or  greater  thickness  can  be  correctly  used. 


Plate  No.  86 


POUL8ON  4  Eger,  Hecla  Architectuhai.  Bronzf  and  Iron  Works,  48  to  51  World  Buildinq,  New  York. 


IRON  STAIRS. 


Iron  Staircases  of  tlie  following  designs,  and  variously  finished,  have  been  built  in  New  York, 
Chicago,  and  most  of  the  principal  cities  of  the  East  and  West.  Iron  balustrade  work  of  these  and 
other  elaborate  designs  is  also  combined  with  wood  and  stone  stairs.  Where  finish  is  desired 
beyond  mere  pai.iting,  advantage  is  taken  of  several  scientific  and  mechanical  processes. 

Electro-plating  is  done  on  stair  work  in  bronze,  brass,  and  antique  brass.  The  railings,  newels, 
and  all  high-relief  lines  and  mouldings  are  polished  to  afford  artistic  contrast.  The  electro-plate  is 
ihcrouglily  substantial  and  durable.  y\ 

Bower-Barff  Rustless  Process. — This  treatment. Imparts  to  the  surface  of  the  iron  a  blue-black 
color  on  rougher  castings,  while  on  polished  work  the  iro'rf ;rfeislfi>§;  its  lustrous  appearance  with  a 
beautiful  steel-blue  shade,  that  harmonizes  well  with  electj-p-plalecj.  v\jt>,ffc  w'hsrc  it  is  used  in  com- 
bination. ;"\'  ' ^  ^f,',-,^^ 

Galvano-plastic. — By  electrolytic  process  are  reproducecf, /wh'en  desired,' {5a'ne'i5,  ,'njbuldings,  and 
ornamental  detail  in  solid  bronz-copper.  These  can  be  given  a  m^dl' ^fiiii<;5i" -of  bronze;  brjfss;  an^tique 
brass,  silver,  oxidized  silver  or  Bower-Barff,  by  a  subsequent  electro-plate.'-.. Th^'-Meqla  Aichi^'tejclural 
Bronze  and  Iron  Works  reproduce  by  this  means  at  a  nominal  cost  details  of  des'igtl  ffom  expensive 
and  highly-artistic  originals — of  which  they  have  a  large  variety — in  solid  metal  tliat  cannot  be 
equaled  by  any  other  known  process.  This  galvano-plastic  work  is  well  adapted  to  panels  in  wain- 
scoting and  decorative  interior  work  ;  and  it  can  be  incorporated  in  the  matter  of  design  with  electro- 
plated cast-iron  work  in  infinitely  varying  ways. 


<    r  •  *  « 


Plate  No. 87 


Plate  No.  88 


POULSON  &  EOEB,  HECLA  ARCHITECTURAL  BRONZE  AND  IRON  WORKS,  48  TO  51  WORLD  BuJLDlNG.  NEW  YORK. 


Plate  No. 90  


P0UL8ON  ir.  Egeh,  Hecla  AHCHiTecruRAi  Bron?f  an&  ikon  Works,  48  to  61  World  Buiidini.,  New  York- 


Plate  No.  91 


PouLSON  &  Eger,  Hecla  Architectural  Bronze  and  Iron  Works,  48  to  51  World  Building,  New  York. 


Plate  No.  92 


Plate  No. 93 


POULSON  &  EGEH,  HECL*  ARCHITECTliHAL  BRON7E  AND  IRON  WORKS,  48  TO  51  WORLD  BUILDING,  NEW  YORK 


Plate  No.  94 


Plate  No  95 


PouLSON  &  Egeb,  Hecla  Architectural  Bronze  and  Iron  Works,  48  to  51  World  Building,  New  York. 


Plate  No.  96 


Po  jtsoN  4.  Eger,  Hecla  ARCHiTECTiiRAi  Bron;f  and  Iron  Works,  48  to  51  World  Builoinq,  New  York. 


Plate  No. 97 


PouLSON  &  Eger,  Hecla  Archjtectural  Bronze  and  Iron  Works,  48  to  51  World  Building,  New  York. 


Plate  No.  98 


PouLSON  &  Eger,  Hecla  AR'^hitectuhal  Bhonjf  »nd  Ihon  Works,  48  To  b1  World  Building,  New  York 


Plate  No. 99 


PouLSON  &  Eger,  Hecla  Architectural  Bronzf  and  Iron  Works,  48  to  51  World  Building,  New  York. 


Platk  No.  100 


J  .  —  

PouLSON  &  Egeh,  Hecl*  AfiCMiTEcrLRAi  Bronze  (^^o  Ikon  WoftKS,  4S  to  b'\  WoiIi-d  Bui^oino,  New  Yoh^ 


Plate 


POULSON  4  EGER,  HECLA  ARCHITECTURAL  BRONiE  AND  IH'JN  WORKS,  48  TO  51  WORLD  BuiLDING,  NEW  YORK. 


Plate  No.  102 


P  L/\rsl 

ScA  LE  |-  I  N.=  1  Ft 


PLAN  OF  IRON  STAIRS: — This  plan  commends  itself  for  many  situations  and  business  purposes 
by  the  very  little  space  occupied;  it  being  also  strong,  compact,  simple,  lasting  and  neat  in  appearance.  In 
ascending  by  this  plan  of  steps  it  is  done  from  side  to  side  as  numbered  1,  2,  3,  etc;  while  a  handrail 
fixed  at  the  middle  is  grasped  as  a  guard  and  assistant.    See  Plate  103  for  a  section  in  elevation  of  this  plan 


Poulson  &  Eger,  Hecia  Architectural  Bronze  and  Iron  Works,  48  to  51  World  Building,  New  York. 


POULSON  &  EGER,  HECLA  ARCHITECTURAL  BRONZE  AND  IRON  WORKS,  48  TO  51  WORLD  BUILDING,  NEW  YORK. 


Plate  No. 104 


VIEW  OF  THREE  DIFFERENT  FORMS  OF  STAIRS  BROUGHT  TOGETHER :— The  flight 
on  the  right  hand  is  a  straight  one;  that  on  the  left  is  in  plan  a  quarter  circle:  the  nriiddle  flight  is 
also  in  plan  a  circular  double  flight  designed  for  convenience  of  ascent  and  descent.  They  are  all 
constructed  after  the  nnethod  shown  and  explained  in  plan  and  section  at  Plates  102,  103.  These 
compact,  neat  and  cleanly  Iron  Stairs  supply  a  want  more  particularly  in  business  places  by  their  great 
economy  of  space. 


PouLSON  &  Eger,  Hecla  Architectural  Bronze  and  Iron  Works,  48  to  51  World  Buildino,  New  York. 


Plate  No.  105. 


Plate  No.  106. 


Plate  No.  1  07. 


Newels  and  Balusters  Manufactured  by  the  Standard  Wooo-TURNiNa  Co.*  206  Greene  St.,  Jersey  City,  N.  J.,  U.  S>  A. 

Send  4  cts.  in  stamps  for  Catalogue  and  Price-List. 


Plate  No.  108. 


Newels  and  Balusters  Manufactured  by  the  Standard  Wood-Turning  Co..  206  Greene  St.,  Jersey  City,  N.  J.,  U.S.A. 

Send  4  cts.  in  stamps  for  Catalogue  and  Price-List. 


Plate  No.  109. 


Newels  and  Balusters  Manufactured  by  the  Stansaro  Wood-Turninq  Co..  206  Greene  St.>  Jersey  City, 

Send  4  cts.  in  stamps  for  Catalogue  and  Price-List. 


MONCKTON'S 

Practical  Geometry 

BEING  A 

SERIES  OF  LESSONS  BEGINNING  WITH  THE  SIMPLEST  PROBLEMS  AND 
IN  THE  COURSE  EMBRACING  ALL  OF  GEOMETRY  LIKELY  TO  BE 
REQUIRED  FOR  THE  USE  OF  EVERY  CLASS  OF  MECHANICS 
OR  THAT  ARE  NEEDED  FOR  INSTRUCTION 
IN  MECHANICAL  SCHOOLS. 


ILLUSTRATED  BY  42  FULL  PAGE  PLATES. 


BV 

JAMES  H.  MONCKTON, 

Author  of  Monckton's  "National  Carpenter  and  Joiner,"  and  Monckton's  "National  Stair  Builder." 
Instructor  for  many  years  of  the  Mechanical  Class  in  "  The  General  Society  of  Mechanics 
and  Tradesmen's  Free  Drawing  School  "  of  the  City  of  New  York. 

Price  $1.00. 

CONTENTS. 


Introduction.  Drawing  instruments,  tools  and  materials  required  to  be- 
gin with,  and  liow  to  use  them.  A  1'  square.  Right  angle  triangles. 
Drawing  boards  and  tacks.  The  compasses  and  its  a  tachments.  The 
line  pen.  Proper  way  to  handle  compasses.  The  lead  pencil.  Six 
rules  for  using  tlie  drawing  instrumenis. 

Plate  i.  Geometry.  A  point.  A  line.  A  curve  line.  A  composite  line. 
A  mixed  curve.  A  zigzag  line.  A  vertical  line.  A  level  line.  A  per 
pendicular. 

Plate  2.  A  circle.  The  radius.  The  diameter.  A  segment  of  a  circle. 
Concentric  circles.    Eccentric  circles.    A  tangent.    Tjngent  circles. 

Plate  3.  To  erect  a  perpendicular  from  a  given  line.  Parallel  lines. 
Oblique  lines.  An  angle.  Curvilinear  angles.  Mixtilinear  ingles. 
To  bisect  a  line. 

Plate  4.  To  bisect  an  angle.  A  diagonal.  To  erect  a  perpendicular  at 
the  end  of  a  given  line.  To  let  fall  a  perpendicular  to  a  given  line 
from  a  given  point.  Angles.  To  trisect  a  quarter  circle.  To  inscribe 
a  square  in  a  circle. 

Plate  5.  A  superficies.  A  plane  figure.  An  equilateral  triangle.  An 
isosceles.  A  scalene  and  a  right  angled  triangle.  A  square.  An 
oblong.  A  rhombus.  Aihomboid.  A  trapezoid.  A  trapezium.  Quad- 
rilaterals and  tetragons. 

Plate  6.  Polygons  and  their  construction.  From  a  given  side  to  con- 
struct an  equilateral  triangle,  a  squ.ire,  a  pentagon,  and  a  hexagon. 

Plate  7.  To  inscribe  an  octagon  within  a  square  (two  methods).  Alti- 
tude or  height  of  a  triangle.  An  angle  nscribed  in  a  semicircle.  To 
inscribe  a  square  within  a  square.  An  equilateral  triangle  inscribed  in 
a  circle.    To  circumscribe  a  square  about  a  circle. 

Plate  8.  To  place  a  line  at  right  angles  to  a  given  line  by  the  use  of  any 
scale  of  equal  parts,  and  its  application  for  a  mechanical  purpose.  To 
circumscribe  an  equilateral  triangle  about  a  given  circle.  To  find  the 
centre  of  a  circle.    To  inscribe  a  circle  in  a  triangle. 

l^LATE  9.  To  copy  an  angle.  To  copy  an  irregular  angular  figure.  To 
describe  a  circle  through  any  three  points  not  in  a  straight  line.  To 
find  a  right  line  equal  to  the  semicircumference  of  a  circle. 

Plate  10.  To  find  the  circumference  of  a  circle  by  another  method.  Con- 
vex and  concave.  The  square  of  the  hypothenuse  of  a  right  angled 
triangle  is  equal  to  the  sum  of  the  other  two  sides. 

Plate  ii.  To  construct  a  triangle  from  given  sides.  To  divide  a  circle 
into  six  equal  parts;  into  eight  equal  parts;  into  twelve  equal  paris. 

Plate  12.  Measurement  of  angles.  Rapid  method  of  dividing  a  gi  en 
space  into  any  number  of  equal  parts. 

Plate  13.    To  divide  a  line  into  any  number  of  equal  parts. 

Plate  14.  Measure  of  the  angle  at  the  centre  and  circumference  of  the 
circle.  To  make  a  square  equal  to  three  given  squares.  To  describe  an 
ellipse  (two  methods). 

Plate  15.   Three  methods  of  describing  the  ellipse. 

Plate  16.  Two  other  methods  of  describing  the  ellipse.  To  draw  a  tan- 
gent to  a  given  point  on  the  ellipse.   To  find  the  point  of  contact. 


Plate  17  To  find  the  axis  of  a  given  ellipse.  To  circumscribe  a  rectan- 
gle by  an  ellipse.  To  find  the  axis  of  an  ellipse  proportional  to  theaxi* 
of  a  given  ellipse.  Parallel  elliptic  lines  impossible.  Tj  describe  an 
approximate  ellipse  with  arcs  of  circles. 

Plate  18.  To  describe  an  approximate  ellipse  more  accurately.  To  de- 
scribe an  eggoid.  To  describe  a  simple  spiral  or  scroll  with  arcs  ol 
circles. 

Plate  iq.  To  describe  an  egg-shaped  oval  with  arcs  of  circles.  To  de 
scribe  a  cycloid.    To  desci  ibe  an  epicycloid. 

Plate  20.  To  describe  an  epicycloid  by  a  circle  rolline;  within  another  cir- 
cle. To  describe  an  involute.  To  describe  a  spiral  or  involute  of  one 
or  more  revolutions. 

Plate  21.    Arches.    A  semicircular  arch.    A  platband.    The  elliptic  arch. 

Plate  22.    To  describe  various  forms  of  gothic  arches. 

Plate  23.  A  rectangular  prism.  The  cube.  A  cube  cut  by  a  plane  pass- 
ing through  the  diagonals.    Cut  by  a  plane  through  the  centre. 

Plate  24.  A  square  prism  cut  at  any  height.  To  develop  the  surface  of 
the  truncated  prism. 

Plate  25.  A  pyramid.  A  triangular  prism.  A  hexagonal  prism.  A  square 
pyramid.    A  hexagonal  pyramid. 

Plate  26.  A  regular  tetrahedron.  A  regular  octahedron.  The  dodecahe- 
dron.  The  icosahedron. 

Plate  27.    A  right  cylinder.    To  find  a  section.    To  find  an  envelope. 

Plate  28.  To  find  the  section  of  a  right  cylinder  cut  obliquely.  To  find 
the  envelope. 

Plate  29.    A  cylindroid     To  find  the  section. 

Plate  30.  To  find  the  section  of  a  cylindroid  cut  obliquely  in  two  direc- 
tions.   A  sphere.    The  covering  of  a  sphere. 

Plate  31  To  describe  the  covering  of  a  sphere  on  lines  parallel  to  centre 
plane.  The  sections  of  a  sphere.  Prolate  spheroid.  Sections  of  a  pro- 
lite  spheroid 

Plate  32.    Covering  of  a  prolate  spheroid.    An  oblate  spheroid.  Sections 

of  an  oblate  spheroid     Covering  of  an  oblate  spheroid. 
Plate  33.    The  helix. 
Plate  34.    The  cone  and  its  sections. 

Plate  35.    The  ellipse  liom  a  cone.    The  parabola.    The  hyperbola. 
Plate  36.   To  describe  the  parabola  by  another  method.  The  ellipse  from 
a  cone. 

Plate  37.  To  describe  a  parabola  by  the  intersection  of  tangent  lines. 
The  hyperbola  on  the  principle  that  sections  of  a  cone  parallel  to  the 
base  are  circles.    To  develop  the  covering  of  a  cone. 

Plate  38.  To  develop  the  covering  of  a  cone,  with  the  trace  of  the  inter- 
section, when  an  hyperbola— also  when  a  parabola— is  produced  by 
cutting  plane. 

Plate  39.    To  develop  the  covering  of  an  oblique  cone  together  with  the 

traces  of  several  cutting  planes. 
Plate  40.    Scale  of  equal  parts.    A  diagonal  decimal  scale. 
Plate  41.   The  protractor.   The  straight  protractor. 


WILLIAM  T.  COMSTOCK,  Publisher,  23  Warren  St.,  New  York. 


- —  -   # 


